A market has both an express checkout line and a super...

A market has both an express checkout line and a super express checkout line. Let X1 denote the number of customers in line at the express checkout at a particular time of day, and let X2 denote the number of customers in line at the super express checkout at the same time. Suppose the joint pmf of X1 and X2 is as given in the accompanying table. (a) What is $P\left(X_{1}=1, X_{2}=1\right),$ that is, the probability that there is exactly one customer in each line? (b) What is $P\left(X_{1}=X_{2}\right),$ that is, the probability that the numbers of customers in the two lines identical? (c) Let A denote the event that there are at least two more customers in one line than in the other line. Express $A$ in terms of $X_{1}$ and $X_{2},$ and calculate the probability of this event. (d) What is the probability that the total number of customers in the two lines is exactly four? At least four? (e) Determine the marginal pmf of $X_{1},$ and then calculate the expected number of customers in line at the express checkout (f) Determine the marginal pmf of $X_{2}$ (g) By inspection of the probabilities $P\left(X_{1}=4\right), P\left(X_{2}=0\right),$ and $P\left(X_{1}=4, X_{2}=0\right),$ are $X_{1}$ and $X_{2}$ independent random variables? Explain.

A market has both an express checkout line and a super express checkout line. Let X1 denote the number of customers in line at the express checkout at a particular time of day, and let X2 denote the number of customers in line at the super express checkout at the same time. Suppose the joint pmf of X1 and X2 is as given in the accompanying table.
(a) What is $P\left(X_{1}=1, X_{2}=1\right),$ that is, the probability that there is exactly one customer in each line?
(b) What is $P\left(X_{1}=X_{2}\right),$ that is, the probability that the numbers of customers in the two lines identical?
(c) Let A denote the event that there are at least two more customers in one line than in the other line. Express $A$ in terms of $X_{1}$ and $X_{2},$ and calculate the probability of this event.
(d) What is the probability that the total number of customers in the two lines is exactly four? At least four?
(e) Determine the marginal pmf of $X_{1},$ and then calculate the expected number of customers in
line at the express checkout
(f) Determine the marginal pmf of $X_{2}$
(g) By inspection of the probabilities $P\left(X_{1}=4\right), P\left(X_{2}=0\right),$ and $P\left(X_{1}=4, X_{2}=0\right),$ are $X_{1}$ and $X_{2}$ independent random variables? Explain.

Key Concepts

Joint Probability Mass Function (pmf)

The joint pmf is a function that gives the probability that a pair of discrete random variables takes on a particular pair of values simultaneously. It is fundamental in describing the relationship between two random variables and must satisfy certain properties such as non-negativity and that the sum of all probabilities equals one.

Marginal Probability Mass Function

The marginal pmf of a random variable is obtained by summing (or integrating) the joint pmf over the range of the other variable. This process effectively 'marginalizes' out the influence of the other variable, allowing for analysis of the probability distribution of a single variable independently from the other.

Expected Value (Expectation)

The expected value of a discrete random variable is a weighted average of all possible values it can take, where each value is weighted by its probability. It is computed by summing the products of the values and their corresponding probabilities, providing a measure of the central tendency of the distribution.

Independence of Random Variables

Two discrete random variables are said to be independent if and only if their joint pmf equals the product of their individual marginal pmfs for all possible values. Independence implies that the occurrence of one variable has no effect on the probability distribution of the other.

Event Representation in Terms of Random Variables

Events in probability problems are often expressed in terms of conditions or inequalities involving random variables. This allows one to calculate probabilities for complex events by summing the appropriate probabilities from the joint or marginal distributions that satisfy the given conditions.

Sum of Random Variables

When dealing with the sum of two discrete random variables, probabilities can be computed for events defined by the sum, such as the total equaling or exceeding a certain value. This involves considering all pairs of outcomes from the joint distribution that contribute to the desired sum, and then accumulating their probabilities.

Transcript

00:01 In this exercise, we are given the joint distribution of the discrete random variables, x1 and x2.

00:08 For part a we are asked for the probability that x1 is equal to 1 and x2 is equal to 1.

00:21 Now we can re -express this as the probability that x1 equals 1 intersected with x2 equals 1.

00:34 So this is the case where both x1 and x2 are equal to 1.

00:40 So that is 0 .15.

00:49 For part b, we're looking for the probability that x1 and x2 are equal.

01:00 So the probability masses where x1 and x2 are equal are these entries along the diagonal.

01:13 You could express this as the probability that x1 equals i and x2 equals i summed over all possible values of i.

01:33 This is only going up to 3 because x2 only goes up to 3.

01:39 So this is equal to 0 .08 plus 0 .15 plus 0 .10 plus 0 .07 to give us a probability of 0 .5.

01:56 For part c, event a is defined as being at least two more customers in one line than in the other.

02:04 And we are asked to express a in terms of x1 and x2 and then calculate the probability of a.

02:12 A is simply the union of x1 being at least 2 greater than x2, with x2 being at least 2 greater than x1.

02:34 And so the probability of a is the sum of the individual probabilities that satisfy either of these conditions.

02:48 So for the first condition, it would be these probabilities.

02:52 I would just group these to keep it organized, plus the probabilities of the second condition, which are these three probabilities, and this all comes out to 0 .22.

03:42 For part d, we are asked for the probability that the total number of customers in the two lines is exactly four, and at least four.

03:53 So for the first one, we're looking for the probability that x1 plus x2 equals four.

04:00 So this is the sum of the probabilities for any situation where x1 plus x2 equals 4.

04:24 So that's when x1 is equal to 1 and x2 is equal to 3, when both of them are equal to 2, when x1 is 3 and x2 is 1, and when x1 is 4 and x2 is 0.

04:55 And this comes out to 0 .17.

05:01 For the probability that x1 plus x2 is at least 4, that's equal to the probability that's probability that x1 plus x2 is equal to 4.

05:16 I'm only breaking it up this way because we've already calculated this probability, plus the probability that x1 plus x2 is at least 5.

05:55 So now i'm just finding all the probabilities where the sum of x1 and x2 is at least 5...

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