In this homework you can use R for any numerical calculations.

Problem 1

  1. Say X is a discrete random variable with \(P(X=k)=c(30-(k-5)^2)\), k=0,..,10, c some constant. Let F be the cdf. Find F(7)

  2. Say X is a continuous random variable with \(P(X=x)=c(30-(x-5)^2), 0<x<10\), 0 otherwise, c some constant. Let F be the cdf. Find F(7).

Problem 2

Let X be an exponential random variable with rate 1, that is it has density \(f(x)=e^{-x},x>0\) and \(f(x)=0\) if \(x<0\). Let Y=X|X>1. Find \(F_Y(2)\). Write a simulation in R that verifies your answer. (R command to generate numbers from this distribution: rexp(n, 1))

Problem 3

Let the random variable X have density \(P(X=k) = k/6, k=1,2,3\). Let the random variable Y have conditional density \(f_{Y|X=k}(y|k)=ke^{-ky}\). Find \(F_Y(2)\).

Problem 4

Say X and Y are two independent continuous random variables with the same density f.ย What can be said about \(P(X<Y)\)?