The lifetimes, in months, of two components in a system, denoted X and Y, have joint probability density function

a. What is the probability that both components last longer than one month?

b. Find the marginal probability density functions fX(0) and fY(y).

c. Are X and Y independent? Explain.

Solution 20E

Step1 of 4:

Let us consider a random variables X and Y they presents the lifetimes, in months, of two components in a system and they have joint probability density function.

Here our goal is:

a).We need to find the probability that both components last longer than one month?

b).We need to find the marginal probability density functions

c).We need to check whether X and Y independent? Explain.

Step2 of 4:

a).

First integrate above equation with respect to “y” we get

Apply integration by parts we get

Apply integral substitution and let u = -2x-y

du = -1dy

)

Substitute “u” value in above equation we get

Now,

Integrate above equation with respect x we get

Apply integral by parts we get

Let us consider u = -2x-1

du = -2

Substitute “u” value in above equation we get

= {}

= 0 - (-)

=

Hence, = .

Step3 of 4:

b).

The marginal probability density function is given by:

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