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  #1  
Old 11-14-2017, 02:39 PM
Nagsinde2002 Nagsinde2002 is offline
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Default Help Needed with these Problems

I am stuck on how to solve these problems. If someone can help me, that would be greatly appreciated.

1) Suppose that X and Y are independent continuous random variables
with density functions fX and fY . Let T = min(X, Y ) and V = max(X, Y ).
Find the marginal density functions fT of T and fV of V.

I know that I should find the CDF of T and V and differentiate those to give me my marginal density functions, but how do I calculate the CDF of T and V?

2) Let 0 < p < 1 and 0 < r < 1 with p not being equal to r. You repeat a trial with success probability p until you see the first success. I repeat a trial with success probability r until I see the first success. All the trials are independent of each other.

a)What is the probability that you and I performed the same number of trials?
b) Let Z be the total number of trials you and I performed altogether. Find the possible values and the probability mass function of Z.

3) Let X and Y be independent exponential random variables with
parameter 1.

(a) Calculate the probability P(Y ≥ X ≥ 2).
(b) Find the density function of the random variable X − Y .

Sorry for this long post, but hopefully someone can help me with these problems.
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  #2  
Old 11-14-2017, 02:41 PM
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NormalDan NormalDan is offline
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Why not show what you've got thus far with your specific questions. People won't just solve them for you.
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  #3  
Old 11-17-2017, 02:47 PM
Z3ta Z3ta is offline
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These look too hard for exam P. Where did you find them?

I will do #2 for you.

Let X be the number of trials until first success (w/ success probability p)
Let Y be the number of trials until first success (w/ success probability r)

(a)


(b) Let be an integer.

unless I made a mistake somewhere...
(PMF is 0 if z is not an integer >=2)
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  #4  
Old 11-17-2017, 03:04 PM
Academic Actuary Academic Actuary is offline
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3a is just a double integral. Draw a picture. For 3b. here is a more general case:

http://www.math.wm.edu/~leemis/chart...ialLaplace.pdf
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