Question

let x be a random variable whose distribution function is f(t)=(t-1)^2 when 1<t<2

Find P(1.5<x <2.5)

Find the expected value of x

Find P(X=E(X))

Find P(X.>E(X))

5. Find the probability that x>E(X) If it is known that x>1.5

Write an instruction in (Basic or any other language) that would generate values of X in a simulation

Any language is fine if you can put the code here please do it.

Thank you !

Answer 1

5.

Let y = F(t) = (t -1)^2

=> (t - 1) = $\sqrt{y}$ (As, t > 1, we assumed $\sqrt{y}$ to be positive)

=>   $t = 1+ \sqrt{y}$

or,

$t = F^{-1}(y) = 1+ \sqrt{y}$

Let y ~ Uniform(0,1). From this, we can generate the values of X in a simulation with below code in R.

On running the code, I got below 10 values of X following the CDF F(t)=(t-1)^2

1.206297 1.453430 1.775084 1.842345 1.779899 1.490227 1.891018 1.978231 1.560255 1.492277