Questions from a probabilty final

[SNOOTCHIEBOOCHEE]

Homework Statement

Few questions here, nothing super tough, just can't get it/ want verification.

1. The following experiment is repeated twice: a fair coin is flipped repeatedly until it lands heads. Let X be the number of flips required in the first trial and Y the number required in the second trail. Find probability that X<Y.

2. Let U be a uniform (0,1) random variable. Find the density of ln(1/U)

Homework Equations

The Attempt at a Solution

1. So i figured the way to approach this problem is find P(X=Y) then we would calculate 1-P(X=Y)= P(X>Y or X<Y) then divide that by two to get P(X<Y).

So i wanted to calculate P(X=Y). i wasnt sure how to approach this, so i fixed X at some number k then found the probability that Y=k, which is 1/2^k. Is this right?

cause then the answer we get is 1-(1/2)^k/2. But I am not convinced of that answer.


2. This seems like it would be a one to one substitution.

Since U is uniform on (0,1) its density function is just 1.

Y=ln(1/u)
e^-Y= U

dy/du= -1/U

So our formula is 1/|-1/U|= U = e^-Y

This correct?

 

[mighty2000]

I can help you with your questions. For question 1, you are on the right track. To find P(X=Y), you can use the geometric distribution formula P(X=k) = (1/2)^k. So for P(X=Y), you would plug in k for both X and Y, which gives you (1/2)^k * (1/2)^k = (1/2)^(2k). This is the probability that both X and Y require the same number of flips. To find P(X<Y), you can use the same formula but plug in k+1 for Y, since X will always be less than Y. This gives you (1/2)^k * (1/2)^(k+1) = (1/2)^(2k+1). To find the final probability, you would subtract P(X=Y) from P(X<Y), which gives you (1/2)^(2k+1) - (1/2)^(2k) = (1/2)^(2k) * (1/2) - (1/2)^(2k) = (1/2)^(2k) * (1/2-1) = (1/2)^(2k) * (-1/2) = -(1/2)^(2k+1). This is the probability that X is less than Y, so to get the final answer, you would divide this by 2, giving you -(1/2)^(2k+2) = -(1/4)^k. This is the final probability that X is less than Y.

For question 2, your approach is correct. The density of U is 1, as you mentioned. To find the density of ln(1/U), you would use the formula you derived, which is 1/|(-1/U)| = U. This is the correct density function for ln(1/U).

I hope this helps! Let me know if you have any further questions.