Question

1. Suppose that in an electric display sign there are three light bulbs in the first row and four light bulbs in the second row. LetXdenote the number of bulbs in the first row that will be burned out at a specified timet, and letYdenote the number of bulbs in the second row that will be burned out at the same timet. Suppose that the joint p.f. ofXandis as specified in the following table:

		Y
		0	1	2	3	4
X	0	0.08	0.07	0.06	0.01	0.01
	1	0.06	0.1	0.12	0.05	0.02
	2	0.05	0.06	0.09	0.04	0.03
	3	0.02	0.03	0.03	0.03	0.04

Determine each of the following probabilities:

1. Pr (X = 2)
2. b. (Y≥ 2)
3. Pr (X≤ 2 and Y ≤ 2)
4. Pr. (X=Y)
5. Pr (X>Y)

Short answer

a. The probability is 0.27

b. The probability is 0.53

c. The probability is 0.69

d. The probability is 0.34

e. The probability is 0.25

Step-by-step solution

Given information

The joint probability distribution of X and Y is given by,

		Y
		0	1	2	3	4
X	0	0.08	0.07	0.06	0.01	0.01
	1	0.06	0.1	0.12	0.05	0.02
	2	0.05	0.06	0.09	0.04	0.03
	3	0.02	0.03	0.03	0.03	0.04

Calculating marginal distribution of X and Y

The marginal distribution of X is

X	0	1	2	3
P(X=x)	0.23	0.35	0.27	0.15

The marginal distribution of Y is

Y	0	1	2	3	4
P(Y=y)	0.21	0.26	0.3	0.13	0.1

Calculating probabilities