Probability doubt

[Noel]

Prove that linear functions of the form y = b+ Bx are normal random vectors provided that x is a normal random vector.
Find E(y) and V(y).
Prove that the normal random variables in y are independent if and only if V(y) is a diagonal matrix

[Noel]

Q.2 Let U be a random variable with the uniform distribution on (0,1) and let X= -c^(-1)*lnU , where c>0.
Show that X has the exponential distribution with scale parameter c.

[Noel]

Q.3 Let X and Y be independent standard normal random variables.
Show that the distribution u of Z = X/Y has the form
u(dz)=dz*(1/(pi*(1+z^2)))
for z€R

[Dreyfus]

I used variable transformation technique.... And first computed the distribution function for X and then differentiated it to get its density function....

[Granpa Simpson]

Great Vaibhav...

[Noel]

Thanks a lot!

[Noel]

Q.4 You are playing a game of coin tossing with Ravi but you suspect that his coin is unfair.A person told you to proceed as follows:toss ravi's coin twice.If the outcome is HT,then call the result "win".If it is TH,then call the result "lose".If it is TT or HH,then ignore the outcome and toss ravi's coin twice again.Follow this procedure till you get either an HT(win) or a TH(lose).Find the probability of winning.