Question

6. Let X Exp( 1/5). Suppose you use X like a random number generator and create 10 numbers based on its density function. Of these 10, let Y be a random variable that counts how many of the 10 numbers are greater than 3. Find E(Y)

Answer 1

$The~p.d.f.~of~X~is\\ f(x)=\left\{\begin{matrix} \frac{1}{5}\exp\left ( -\frac{x}{5} \right ),~~x>0\\ 0~otherwise~~~~~~~~~~ \end{matrix}\right.\\ p=P(X>3)=\int_{3}^{\infty}\frac{1}{5}\exp\left ( -\frac{x}{5} \right )dx=\left [ - \exp\left ( -\frac{x}{5} \right )\right ]^\infty_3= \exp\left ( -\frac{3}{5} \right )\\ Y=r.v.~denoting~the~counts~how~many~of~the~10~no.s.~are~greater~than~3.\\ Then~Y\sim Binomial(10,p).\\ E(Y)=10p=10 \exp\left ( -\frac{3}{5} \right )=5.49\approx 5$