Question

8. A random variable X has an exponential distribution with mean 6. Then P [(X>9)/(X>3)] is
a) e/3
b) 3/e
c) e-1
d) e-3​

Answer 1

Given,

Random variable X has an exponential distribution

E(X)= 6

To Find,

$P[\frac{(X > 9)}{(X > 3)}]$

Solution,

In case of exponential distibution

E(X)= 1/λ

6 = 1/λ

λ = $\frac{1}{6}$

Now,

$P[\frac{(X > 9)}{(X > 3)}]$ =$\frac{P[(X > 9) intersection (X > 3) ]}{P(X > 3)}$

              = $\frac{P(X > 9)}{P(X > 3)}$

{ now using, P(X≥x) = e^-λx }

              = $\frac{e^{\frac{-1}{6}*9 } }{e^{\frac{-1}{6}*3 } }$

              = $\frac{e^{\frac{-3}{2} } }{e^{-\frac{1}{2} } }$

              = $e^{-1}$

Hence, the required answer is $e^{-1}$.