# Actuarial Science help: Survival Models

• Jan 16th 2010, 05:33 PM
Olga
Actuarial Science help: Survival Models

Let X1 and X2 be independent random variables. Define the variables Y =min(X1,X2) and Z = max(X1,X2).
a) Show that Sy(y) is the product of the survival distribution functions of X1 and X2.
b) Show that Fz(z) is the cumulative distribution functions of X1 and X2.
c)Show that if X1 and X2 both have exponential distributions, then Y also has an exponential distribution, but Z does not.
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Let the independent random variables X1 and X2 both have exponential distributions, with parameters lambda1 and lambda2 respectively, where lambda1>lambda2. Let Y =min(X1,X2) and Z =max(X1,X2). Given that
Sy(2) = 0.24 and Sz(2) = 0.86, find the value of lambda1.
• Jan 19th 2010, 05:16 PM
matheagle
First of all, I do not use this S function.
So I had to guess it was 1-F.

$S_Y(y)=P(Y>y) =P(X_1>y, X_2>y) =P(X_1>y)P(X_2>y)= S_{X_1}(y)S_{X_2}(y)$
• Jan 21st 2010, 08:53 AM
Olga
Thank you for your help. Your explanation helped me figure out how to do the rest of the problems as well.