Probability Distributions

Random Variables and Probability Distributions

see equation (2.1)

Expectation of Random Variables

y<-rbinom(10000000,3,0.50) # simulate 10000000 times the no.ofsuccesses in 3 trials
head(y,10) # the first 10 of the 10 million observations generated

 [1] 3 2 2 0 2 3 1 0 1 2

mean(y);sd(y)

[1] 1.500123

[1] 0.8658677

y<-runif(10000000, min=0, max=100) # runif simulates uniform random variable
head(y,5) # first 5 of ten million generated (to 6 decimal places)

[1] 80.90635 36.73495 12.70536 17.41020 71.02995

mean(y);sd(y) # mean and standard deviation of ten million simulated uniform rv's

[1] 49.99301

[1] 28.86167

Exercises

Compute the expectation and variance of \(X\) in terms of \(Y\).

Compute the expectation and variance of \(Z\).