Final

Problem 1
Say the rv X has density f(x)=c·√x, 0<x<1, 0 otherwise.

a) find c
b) find P(X>0.5)
c) find the mean and the standard deviation of X

Problem 2
Say X is a rv with P(X=x)=|x|/110 for x=0,±1, .., ±10. Let Y=X²/100. Find the pmf of Y. Find E[Y]

Problem 3
Say the rv (X,Y) has joint density f(x,y)=2.943/(x+y), 1≤x,y≤2. Let U=X²Y and V=XY. Find the joint density of (U,V)

Problem 4
Say the rv (X,Y) has joint density f(x,y)=2.943/(x+y), 1≤x,y≤2. Find E[X|Y=y]

Problem 5
The discrete rv (X,Y) has a joint pmf given by

so for example P(X=3,Y=1)=0.5-q. Show that X and Y are uncorrelated if and only if q=¼+p

Problem 6
Each customer who enters a certain store will buy something with probability p, independent of any other customer. The number of customers entering the store during the day has a Poisson distribution with mean l. Show that the probability of no sale during a whole day is e^-lp