The purpose of this section is to study two basic types of objects that form part of the model of a random experiment.
The sample space of a random experiment is a set \(S\) that includes all possible outcomes of the experiment; the sample space plays the role of the universal set when modeling the experiment. For simple experiments, the sample space may be precisely the set of possible outcomes. More often, for complex experiments, the sample space is a mathematically convenient set that includes the possible outcomes and perhaps other elements as well. For example, if the experiment is to throw a standard die and record the outcome, the sample space is \(S = \{1, 2, 3, 4, 5, 6\}\), the set of possible outcomes. On the other hand, if the experiment is to capture a cicada and measure its body weight (in milligrams), we might conveniently take the sample space to be \(S = [0, \infty)\), even though most elements of this set are practically impossible (we hope!).
Often the outcome of a random experiment consists of one or more real measurements, and thus, the sample space consists of all possible measurement sequences. Thus, in many cases, the sample space of a random experiment is a subset of \(\R^n\) for some \(n\).
If we have \(n\) experiments with sample spaces \(S_1, S_2, \ldots, S_n\), then the Cartesian product \(S_1 \times S_2 \times \cdots \times S_n\) is the natural sample space for the compound experiment that consists of performing the \(n\) experiments in sequence. In particular, if we have a basic experiment with sample space \(S\), then \(S^n\) is the natural sample space for the compound experiment that consists of \(n\) replications of the basic experiment. Similarly, if we have an infinite sequence of experiments with sample spaces \((S_1, S_2, \ldots)\) then \(S_1 \times S_2 \times \cdots\) is the natural sample space for the compound experiment that consists of performing the given experiments in sequence. In particular, the sample space for the compound experiment that consists of indefinite replications of a basic experiment is \(S^\infty = S \times S \times \cdots \). This is an essential special case, because (classical) probability theory is based on the idea of replicating a given experiment.
Certain subsets of the sample space of an experiment are referred to as events. Thus, an event is a set of outcomes of the experiment. Each time the experiment is run, a given event \(A\) either occurs, if the outcome of the experiment is an element of \(A\), or does not occur, if the outcome of the experiment is not an element of \(A\). Intuitively, you should think of an event as a meaningful statement about the experiment.
In the section on Measure Theory, we discuss some technical conditions that must be imposed on the collection of events. These need not concern you if you are new to the study of probability; we will simply assume that all subsets of the sample space that we mention are valid events.
In particular, the sample space \(S\) itself is an event; by definition it always occurs. At the other extreme, the empty set \(\emptyset\) is also an event; by definition it never occurs.
The standard algebra of sets leads to a grammar for discussing random experiments and allows us to construct new events from given events. In the following exercises, suppose that \(A\) and \(B\) are events.
\(A \subseteq B\) if and only if the occurrence of \(A\) implies the occurrence of \(B\).
\(A \cup B\) is the event that occurs if and only if \(A\) occurs or \(B\) occurs.
\(A \cap B\) is the event that occurs if and only if \(A\) occurs and \(B\) occurs.
\(A\) and \(B\) are disjoint if and only if they are mutually exclusive; they cannot both occur on the same run of the experiment.
\(A \setminus B\) is the event that occurs if and only if \(A\) occurs and \(B\) does not occur.
\(A^c\) is the event that occurs if and only if \(A\) does not occur.
\((A \cap B^c) \cup (B \cap A^c)\) is the event that occurs if and only if one but not both of the given events occurs. Recall that this event is the symmetric difference of \(A\) and \(B\), and is somtimes denoted \(A \Delta B\).
\((A \cap B) \cup (A^c \cap B^c)\) is the event that occurs if and only if both or neither of the given events occurs.
In the Venn diagram applet, observe the diagram of each of the 16 events that can be constructed from \(A\) and \(B\).
Suppose now that \(\mathscr{A} = \{A_i: i \in I\}\) is a collection of events for the random experiment, where \(I\) is a countable index set.
\( \bigcup \mathscr{A} = \bigcup_{i \in I} A_i \) is the event that occurs if and only if at least one event in the collection occurs.
\( \bigcap \mathscr{A} = \bigcap_{i \in I} A_i \) is the event that occurs if and only if every event in the collection occurs:
\(\mathscr{A}\) is a pairwise disjoint collection if and only if the events are mutually exclusive; at most one of the events could occur on a given run of the experiment.
Suppose now that \((A_1, A_2, \ldots\)) is a countably infinite sequence of events.
\(\bigcap_{n=1}^\infty \bigcup_{i=n}^\infty A_i\) is the event that occurs if and only if infinitely many of the given events occur. This event is sometimes called the limit superior of \((A_1, A_2, \ldots)\).
\(\bigcup_{n=1}^\infty \bigcap_{i=n}^\infty A_i\) is the event that occurs if and only if all but finitely many of the given events occur. This event is sometimes called the limit inferior of \((A_1, A_2, \ldots)\).
Limit superiors and inferiors are discussed in more detail in the section on convergence.
Suppose again that we have a random experiment with sample space \(S\). A function \(X\) from \(S\) into another set \(T\) is called a (\(T\)-valued) random variable. Probability has its own notation, very different from other branches of mathematics. As a case in point, random variables are usually denoted by capital letters near the end of the alphabet.
In the section on Measure Theory, we will discuss a technical condition that must be imposed on random variables (and functions generally). These need not concern you if you are new to the study of probability; we will simply assume that all functions we mention are admissible.
Intuitively, you should think of a random variable \(X\) as a measurement of interest in the context of the random experiment. A random variable \(X\) is random in the sense that its value depends on the outcome of the experiment, which cannot be predicted with certainty before the experiment is run. Each time the experiment is run, an outcome \(s \in S\) occurs, and a given random variable \(X\) takes on the value \(X(s)\). In general, as you will see, the notation of probability suppresses references to the sample space. Indeed, sometimes the sample space is hidden in the sense that we don't know what it is.
Often, a random variable takes values in a subset \(T\) of \(\R^k\) for some \(k\). If \(k \gt 1\) then we write \(\bs{X} = (X_1, X_2, \ldots, X_k)\) where \(X_i\) is a real-valued random variable for each \(i\). In this case, we usually refer to \(\bs{X}\) as a random vector, to emphasize its higher-dimensional character. A random variable can have an even more complicated structure. For example, if the experiment is to select \(n\) objects from a population and record various real measurements for each object, then the outcome of the experiment is a vector of vectors: \(\bs{X} = (X_1, X_2, \ldots, X_k)\) where \(X_i\) is the vector of measurements for the \(i\)th object. There are other possibilities; a random variable could be an infinite sequence, or could be set-valued. Specific examples are given in the computational exercises below. However, the important point is simply that a random variable is a function defined on the sample space \(S\).
If \(X\) is a random variable taking values in \(T\), and \(B \subseteq T\), then we use the more suggestive notation for the inverse image:
\[ \{X \in B\} = \{s \in S: X(s) \in B\} \]Note that this is an event (a subset of \(S\)). In words, a statement about the random variable defines an event.
A special case of this notation is
\[ \{X = x\} = \{s \in S: X(s) = x\}, \quad x \in T \]Suppose in particular that \(S\) is a real-valued random variable for an experiment. Another special cases of our notation is
\[ \{ a \leq X \leq b\} = \{ X \in [a, b]\} = \{s \in S: a \leq X(s) \leq b\} \]Of course, there are analogous results for other types of inequalities. The following exercise is a restatement of the fact that inverse images of a function preserve the set operations; only the notation changes (and is simpler).
Suppose that \(X\) is a random variable taking values in \(T\), and that \(A\) and \(B\) are subsets of \(T\).
As with a general function, the result in part (a) hold for the union of a countable collection of subsets, and the result in part (b) hold for the intersection of a countable collection of subsets. No new ideas are involved; only the notation is more complicated.
Suppose again that we have a random experiment with sample space \(S\). The outcome of the experiment itself can be thought of as a random variable. Specifically, let \(X\) denote the identify function on \(S\) so that \(X(s) = s\) for \(s \in S\). Then trivially \(X\) is a random variable, and the events that can be defined in terms of \(X\) are simply the original events of the experiment:
\[ \{ X \in A\} = A, \quad A \subseteq S \]If \(Y\) is another random variable for the experiment, taking values in a set \(T\), then \(Y\) is a function of \(X\). That is, there is a function \(g\) from \(S\) into \(T\) such that \(Y\) is the composition of \(g\) with \(X\). We use the more descriptive notation \(g(X)\) instead of \(g \circ X\). Thus,
\[ Y(s) = g[X(s)], \quad s \in S \]We could refer to \(X\) as the outcome variable and \(Y\) as a derived variable. In many problems of elementary probability theory, the basic object of interest is a random variable \(X\). Whether \(X\) is the basic outcome variable or a derived variable is often irrelevant.
For an event \(A\), the indicator function of \(A\) is called the indicator variable of \(A\). The value of this random variables tells us whether or not \(A\) has occurred:
\[ \bs{1}_A = \begin{cases} 1, & A \text{ occurs} \\ 0, & A \text{ does not occur} \end{cases} \]That is, as a function on the sample space,
\[ \bs{1}_A(s) = \begin{cases} 1, & s \in A \\ 0, & s \notin A \end{cases} \]If \(X\) is a random variable that takes values 0 and 1, then \(X\) is the indicator variable of the event \(\{X = 1\}\).
Recall also that the set algebra of events translates into the arithmetic algebra of indicator variables.
Suppose that \(A\) and \(B\) are events.
The results in part (a) extends to arbitrary intersections and the results in part (b) extends to arbitrary unions.
Recall that probability theory is often illustrated using simple devices from games of chance: coins, dice, cards, spinners, urns with balls, and so forth. Examples based on such devices are pedagogically valuable because of their simplicity and conceptual clarity. On the other hand, remember that probability is not only about gambling and games of chance. Rather, try to see problems involving coins, dice, etc. as metaphors for more complex and realistic problems.
The basic coin experiment consists of tossing a coin \(n\) times and recording the sequence of scores \((X_1, X_2, \ldots, X_n)\) (where 1 denotes heads and 0 denotes tails). This experiment is a generic example of \(n\) Bernoulli trials, named for Jacob Bernoulli.
Consider the coin experiment with \(n = 4\), and Let \(Y\) denote the number of heads.
In the simulation of the coin experiment, set \(n = 4\). Run the experiment 100 times and count the number of times that the event \(\{Y = 2\}\) occurs.
Now consider the general coin experiment with the coin tossed \(n\) times, and let \(Y\) denote the number of heads.
The basic dice experiment consists of throwing \(n\) distinct \(k\)-sided dice (with sides numbered from 1 to \(k\)) and recording the sequence of scores \((X_1, X_2, \ldots, X_n)\). This experiment is a generic example of \(n\) multinomial trials. The special case \(k = 6\) corresponds to standard dice.
Consider the dice experiment with \(n = 2\) standard dice. Let \(S\) denote the sample space, \(A\) the event that the first die score is 1, and \(B\) the event that the sum of the scores is 7. Give each of the following events in list form:
In the simulation of the dice experiment, set \(n = 2\). Run the experiment 100 times and count the number of times each event in the previous exercise occurs.
Consider the dice experiment with \(n = 2\) standard dice, and let \(Y\) denote the sum of the scores, \(U\) the minimum score, and \(V\) the maximum score.
Consider again the dice experiment with \(n = 2\) standard dice, and let \(Y\) denote the sum of the scores, \(U\) the minimum score, and \(V\) the maximum score. Give each of the following as subsets of the sample space, in list form.
In the dice experiment, set \(n = 2\). Run the experiment 100 times. Count the number of times each event in the previous exercise occurred.
In the general dice experiment with \(n\) distinct \(k\)-sided dice, let \(Y\) denote the sum of the scores, \(U\) the minimum score, and \(V\) the maximum score.
An experiment consists of throwing a pair of standard dice repeatedly until the sum of the two scores is either 5 or 7. Let \(A\) denote the event that the sum is 5 rather than 7 on the final throw. Experiments of this type arise in the casino game craps.
Suppose that 3 standard dice are rolled and the sequence of scores \((X_1, X_2, X_3)\) is recorded. A person pays $1 to play and then receives $1 for each die that lands on the number 6. Let \(W\) denote the person's net winnings. This is the game of chuck-a-luck and is treated in more detail in the chapter on Games of Chance.
In the die-coin experiment, a standard die is rolled and then a coin is tossed the number of times shown on the die. The sequence of coin scores \(\bs{X}\) is recorded (0 for tails, 1 for heads). Let \(N\) denote the die score and \(Y\) the number of heads.
Run the simulation of the die-coin experiment 10 times. For each run, give the values of the random variables \(\bs{X}\), \(N\), and \(Y\) of the previous exercise. Count the number of times the event \(\{Y = 2\}\) occurs.
In the coin-die experiment, we have a coin and two distinct dice, say one red and one green. First the coin is tossed, and then if the result is heads the red die is thrown, while if the result is tails the green die is thrown. The coin score \(X\) and the score of the chosen die \(Y\) are recorded. Suppose now that the red die is a standard 6-sided die, and the green die a 4-sided die.
Run the coin-die experiment 100 times, with the a fair 6-sided die and a fair 4-sided die. Count the number of times that event \(\{Y \ge 3\}\) occurs.
Recall that many random experiments can be thought of as sampling experiments. For the general finite sampling model, we start with a population \(D\) with \(m\) (distinct) objects. We select a sample of \(n\) objects from the population. If the sampling is done in a random way, then we have a random experiment with the sample as the basic outcome. Thus, the sample space of the experiment is literally the space of samples; this is the historical origin of the term sample space. There are four common types of sampling from a finite population, based on the criteria of order and replacement. Recall the following facts from the section on combinatorial structures:
If we sample with replacement, the sample size \(n\) can be any positive integer. If we sample without replacement, the sample size cannot exceed the population size, so we must have \(n \in \{1, 2, \ldots, m\}\).
The basic coin and dice experiments are examples of sampling with replacement. If we toss a coin \(n\) times and record the sequence of scores (where as usual, 0 denotes tails and 1 denotes heads), then we generate an ordered sample of size \(n\) with replacement from the population \(\{0, 1\}\). If we throw \(n\) (distinct) standard dice and record the sequence of scores, then we generate an ordered sample of size \(n\) with replacement from the population \(\{1, 2, 3, 4, 5, 6\}\).
Suppose that the sampling is without replacement (the most common case). If we record the ordered sample \(\bs{X} = (X_1, X_2, \ldots, X_n)\), then the unordered sample \(\bs{W} = \{X_1, X_2, \ldots, X_n\}\) is a random variable (that is, a function of \(\bs{X}\)). On the other hand, if we just record the unordered sample \(\bs{W}\) in the first place, then we cannot recover the ordered sample. Note also that the number of ordered samples of size \(n\) is simply \(n!\) times the number of unordered samples of size \(n\). No such simple relationship exists when the sampling is with replacement. This will turn out to be an important point when we study probability models based on random samples, in the next section.
Consider a sample of size \(n = 3\) chosen without replacement from the population \(\{a, b, c, d, e\}\).
Traditionally in probability theory, an urn containing balls is often used as a metaphor for a finite population.
Suppose that an urn contains 50 (distinct) balls. A sample of 10 balls is selected from the urn. Find the number of samples in each of the following cases:
Suppose again that we have a population \(D\) with \(m\) (distinct) objects, but suppose now that each object is one of two types--either type 1 or type 0. Such populations are said to be dichotomous. Here are some specific examples:
Suppose that the population \(D\) has \(r\) type 1 objects and hence \(m - r\) type 0 objects. Of course, we must have \(r \in \{0, 1, \ldots, r\}\). Now suppose that we select a sample of size \(n\) without replacement from the population. Note that this model has three parameters: the population size \(m\), the number of type 1 objects in the population \(r\), and the sample size \(n\).
Let \(Y\) denote the number of type 1 objects in the sample, a random variable for the experiment.
A batch of 50 components consists of 40 good components and 10 defective components. A sample of 5 components is selected, without replacement. Let \(Y\) denote the number of defectives in the sample.
Run the simulation of the ball and urn experiment 100 times for the parameter values in the last exercise. Note the values of the random variable \(Y\).
Recall that a standard card deck can be modeled by the product set
\[D = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, j, q, k\} \times \{\clubsuit, \diamondsuit, \heartsuit, \spadesuit\}\]where the first coordinate encodes the denomination or kind (ace, 2-10, jack, queen, king) and where the second coordinate encodes the suit (clubs, diamonds, hearts, spades). Sometimes we represent a card as a string rather than an ordered pair (for example \(q \heartsuit\) rather than \((q, \heartsuit)\)).
Card games involve sampling without replacement from the deck \(D\), which plays the role of the population. Thus, the basic card experiment consists of dealing \(n\) cards from a standard deck without replacement; in this special context, the sample of cards is often referred to as a hand. Just as in the general sampling model, if we record the ordered hand \(\bs{X} = (X_1, X_2, \ldots, X_n)\), then the unordered hand \(\bs{W} = \{X_1, X_2, \ldots, X_n\}\) is a random variable (that is, a function of \(\bs{X}\)). On the other hand, if we just record the unordered hand \(\bs{W}\) in the first place, then we cannot recover the ordered hand. Finally, recall that \(n = 5\) is the poker experiment and \(n = 13\) is the bridge experiment. The game of poker is treated in more detail in the chapter on Games of Chance.
Suppose that a single card is dealt from a standard deck. Let \(Q\) denote the event that the card is a queen and \(H\) the event that the card is a heart. Give each of the following events in list form:
In the card experiment, set \(n = 1\). Run the experiment 100 times and count the number of times each event in the previous exercise occurs.
Suppose that two cards are dealt from a standard deck and the sequence of cards recorded. Let \(Q_i\) denote the event that the \(i\)th card is a queen and \(H_i\) the event that the \(i\)th card is a heart, for \(i \in \{1, 2\}\). Find the number of outcomes in each of the following events:
Consider the general card experiment in which \(n\) cards are dealt from a standard deck, and the ordered hand \(\bs{X}\) is recorded.
Consider the bridge experiment of dealing 13 cards from a deck and recording the unordered hand. In the most common point counting system, an ace is worth 4 points, a king 3 points, a queen 2 points, and a jack 1 point. The other cards are worth 0 points. Let \(V\) denote the point value of the hand.
In the card experiment, set \(n = 13\) and run the experiment 100 times. For each run, compute the value of each of the random variable \(V\) in the previous exercise.
Consider the poker experiment of dealing 5 cards from a deck and recording the unordered hand. Find the cardinality of each of the events below, as a subset of the sample space.
Run the poker experiment 1000 times. Note the number of times that the events \(A\), \(B\), and \(C\) in the previous exercise occurred.
Consider the bridge experiment of dealing 13 cards from a standard deck, and recording the unordered hand. Let \(Y\) denote the number of hearts in the hand and let \(Z\) denote the number of queens in the hand.
In the experiments that we have considered so far, the sample spaces have all been discrete (that is, either finite or countably infinite). Aside from their historical interest, Buffon's experiments, named for Compte de Buffon are examples of random experiments in which the outcome variables are continuous.
Buffon's coin experiment consists of tossing a coin with radius \(r \le \frac{1}{2}\) randomly on a floor covered with square tiles of side length 1. The coordinates \((X, Y)\) of the center of the coin are recorded relative to axes through the center of the square in which the coin lands.
In Buffon's coin experiment, let \(A\) denote the event that the coin does not touch the sides of the square and let \(Z\) denote the distance form the center of the coin to the center of the square.
Run Buffon's coin experiment 100 times with \(r = 0.2\). For each run, note whether event \(A\) occurs and compute the value of random variable \(Z\).
In a simple model of structural reliability, a system is composed of \(n\) components, each of which is either working or failed. The state of component \(i\) is an indicator random variable \(X_i\), where 1 means working and 0 means failure. Thus, \(\bs{X} = (X_1, X_2, \ldots, X_n)\) is a vector of indicator random variables that specifies the states of all of the components, and therefore the sample space of the experiment is \(\{0, 1\}^n\). The system as a whole is also either working or failed, depending only on the states of the components and how the components are connected together. Thus, the state of the system is an indicator random variable defined on this state space: \(Y(\bs{X})\). The state of the system (working or failed) as a function of the states of the components is the structure function.
A series system is working if and only if each component is working. The state of the system is
\[U = X_1 X_2 \cdots X_n = \min\{X_1, X_2, \ldots, X_n\}\]A parallel system is working if and only if at least one component is working. The state of the system is
\[V = 1 - (1 - X_1)(1 - X_2) \cdots (1 - X_n) = \max\{X_1, X_2, \ldots, X_n\}\]More generally, a \(k\) out of \(n\) system is working if and only if at least \(k\) of the \(n\) components are working. Note that a parallel system is a 1 out of \(n\) system and a series system is an \(n\) out of \(n\) system. A \(k\) out of \(2 k\) system is a majority rules system.
The state of the \(k\) out of \(n\) system is given by the formula below. The structure function can also be expressed as a polynomial in the variables.
\[U_{n,k} = 1 \text{ if and only if } \sum_{i = 1}^n X_i \ge k\]Explicitly give the state of the \(k\) out of 3 system, as a polynomial function of the component states \((X_1, X_2, X_3)\), for each \(i \in \{1, 2, 3\}\).
In some cases, the system can be represented as a graph or network. The edges represent the components and the vertices the connections between the components. The system functions if and only if there is a working path between two designated vertices, which we will denote by \(a\) and \(b\).
Find the state of the Wheatstone bridge network shown below, as a function of the component states. The network is named for Charles Wheatstone.
Note that if component 3 works, then the system works if and only if component 1 or 2 works, and component 4 or 5 works. On the other hand, if component 3 does not work, then the system works if and only if components 1 and 4 work, or components 2 and 5 work.
Not every function \(s\) from \(\{0, 1\}^n\) into \(\{0, 1\}\) makes sense as a structure function. Explain why the following properties might be desirable:
The model just discussed is a static model. We can extend it to a dynamic model by assuming that component \(i\) is initially working, but has a random time to failure \(T_i\), taking values in \([0, \infty)\), for each \(i \in \{1, 2, \ldots, n\}\). Thus, the basic outcome of the experiment is the random vector of failure times: \((T_1, T_2, \ldots, T_n)\) and the sample space of the experiment is \([0, \infty)^n\).
Consider the dynamic reliability model for a system with structure function \(s\) (valid in the sense of Exercise 53).
Suppose that we have two devices and that we record \((X, Y)\), where \(X\) is the failure time of device 1 and \(Y\) is the failure time of device 2. Both variables take values in the interval \([0, \infty)\), where the units are in hundreds of hours. Sketch each of the following events:
Consider again a simple model for an inherited trait, say the color of pods in a pea plant, either green or yellow. There are two genes for pod color, one from each parent, so the possible genotypes are
The green gene is dominant, so a plant with genotype \(gg\) or \(gy\) will have green pods while a plant with genotype \(yy\) will have yellow pods.
Suppose that \(n\) (distinct) pea plants are collected and the sequence of pod color genotypes is recorded.
Next, let's recall that a sex-linked hereditary disorder is a disorder caused by a defective gene on the \(X\) chromosome, one of the chromosomes that determine gender. We will let \(h\) denote the healthy gene and \(d\) the defective gene. Women have two \(X\) chromosomes, and the normal gene is dominant. Thus a woman with genotype \(hh\) is completely normal with respect to the disorder; a woman with genotype \(hd\) is healthy, but is a carrier since she can pass the defective gene to her children; and a woman with genotype \(dd\) has the disorder. Men have only one \(X\) chromosome; the other male chromosome \(Y\) typically plays no role in the disease. Thus, a man with genotype \(h\) is normal, while a man with genotype \(d\) has the disorder.
Suppose that \(n\) women are sampled and the sequence of genotypes is recorded.
The emission of elementary particles from a sample of radioactive material occurs in a random way. Suppose that the time of emission of the \(i\)th particle is a random variable \(T_i\) taking values in \((0, \infty)\). If we measure these arrival times, then basic outcome vector is \((T_1, T_2, \ldots)\) and the sample space of the experiment is \(S = \{(t_1, t_2, \ldots): 0 \lt t_1 \lt t_2 \lt \cdots\}\).
Run the simulation of the gamma experiment in single-step mode for different values of the parameters. Observe the arrival times.
Now let \(N_t\) denote the number of emissions in the interval \((0, t]\). Then
Run the simulation of the Poisson experiment in single-step mode for different parameter values. Observe the arrivals in the specified time interval.
In the basic cicada experiment, a cicada in the Middle Tennessee area is captured and the following measurements recorded: body weight (in grams), wing length, wing width, and body length (in millimeters), species type, and gender. The cicada data set gives the results of 104 repetitions of this experiment.
In the basic M&M experiment, a bag of M&Ms (of a specified size) is purchased and the following measurements recorded: the number of red, green, blue, yellow, orange, and brown candies, and the net weight (in grams). The M&M data set gives the results of 30 repetitions of this experiment.