probabiltiy5

 typed answer with detail explanations in pdf file

 strict due.

 If you think you can't do some of questions, just don't start my work please.

 #8 and #9 don't need to be solved. I just want 1-7 to be solved.

 For number 6, you may want to use this code.

 You can install this R program here: http://www.r-project.org/index.html

 # pdf will contain the exact pdf
x <- seq(-4,4,0.01)
pdf=exp(-x*x/2)/(sqrt(2*3.1416))

# number of samples we will generate, mean and stand dev for normal:
nsamples=100000
mean=0
sd=1

# the vector samples will contain the samples of the normal RV
samples <- rnorm(nsamples,mean,sd)

# plot a histogram of the samples using bins of with 0.2
hist(samples,freq=F,ylim=c(0,0.5),breaks=c(seq(-5.,5.,0.2)))

# plot the pdf in red
lines(x,pdf,col="red",lwd=2)