Question

Let X denote the number of times (1, 2, or 3 times) a certain machine malfunctions on any given day. Let Y denote the number of times (1, 2, or 3 times) a technician is called on an emergency call. The joint probability distribution fxy(x, y) is given by 1 3 0.05 2 0.05 0.05 0 0.1 0.1 0.35 0.1 0.2(a) Evaluate the marginal pdf and the mean of X,(b) Evaluate the marginal pdf and the mean of Y.(c) Evaluate P(Y = 3|X = 2).(d) Evaluate E(XY) and determine if X and Y are uncorrelated.

Answer 1

To find the marginal pdf and mean of X and Y, sum the probabilities in each column or row of the joint probability distribution table, respectively. To find P(Y = 3|X = 2), locate the cell in the joint probability distribution table that corresponds to X = 2 and Y = 3. To calculate E(XY) and determine if X and Y are uncorrelated, multiply each pair of X and Y values by their joint probability and compare the calculated E(XY) with the product of the mean values of X and Y.

Step-by-step explanation:

(a) Marginal pdf and mean of X:

• To find the marginal pdf of X, we need to sum the probabilities of X for each value of Y.
• The marginal pdf of X is obtained by summing the probabilities in each column of the joint probability distribution table.
• The mean of X can be calculated by multiplying each value of X by its probability and summing the products.

(b) Marginal pdf and mean of Y:

• To find the marginal pdf of Y, we need to sum the probabilities of Y for each value of X.
• The marginal pdf of Y is obtained by summing the probabilities in each row of the joint probability distribution table.
• The mean of Y can be calculated by multiplying each value of Y by its probability and summing the products.

(c) P(Y = 3|X = 2):

• To find P(Y = 3|X = 2), we need to locate the cell in the joint probability distribution table that corresponds to X = 2 and Y = 3.
• The value in that cell represents the conditional probability of Y = 3 given X = 2.

(d) E(XY) and correlation:

• E(XY) can be calculated by multiplying each pair of X and Y values by their joint probability and summing the products.
• To determine if X and Y are uncorrelated, we can compare the calculated E(XY) with the product of the mean values of X and Y.