Homework 1

Problem 1

Let the rv X have the probability mass function f(x;λ) = c(λ+x), x=1,2,..,N, where N is known, c is a constant and λ is an unknown parameter.

1. find c
2. find E[X]

Problem 2

Let the rv (X,Y) be the uniform distribution , on the area 0<x<y^2<1, 0<y<1

1. find cor(X,Y)
2. find E[X|Y=y]

Problem 3

Say we have discrete random variables \(X_1,..,X_{50}\), independent, with \(P(X_i=x)=c(1+(x-2)^2)\), \(x=0,1,2,3,4\). Find (an approximate value of)

\[P(X_1+...+X_{50}>110)\]

Problem 4

Let \(X\sim Exp(1)\) and \(Y\sim Exp(2)\), independent. Find \(P(X+Y<1)\).