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Science Forum Index » Statistics - Math Forum » problem relating to conditional probability
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Message |
| Guest |
 Posted: Wed Apr 16, 2008 6:53 am |
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Dear forumers,
I am working on the problem below. I will appreciate your comments and
your help.
X is generated from Y with probability 0.5 and from Z with probability
0.5.
Y~exp(lambda) with pdf lambda*exp(-lambda*y)
Z~exp(theta) with pdf theta*exp(-theta*z)
Questions: What is the survival function of X? What is the hazard
function of X?
Define random variable V such that V=0 if X=Y and V=1 if X=Z
S(x)
=P(X>x)
=E{P(X>x)| V}
=P(X>x)*P{V=0} + P(X>x)*P{V=1}
=P(X>x)*P{X=Y} + P(X>x)*P{X=Z}
=P(Y>x)*0.5 + P(Z>x)*0.5
=0.5*exp(-lambda*x) + 0.5*exp(-theta*x)
f(x)=-dS(x)/dx
=0.5*lambda*exp(-lambda*x) + 0.5*theta*exp(-theta*x)
h(x)=f(x)/S(x)
={0.5*lambda*exp(-lambda*x) + 0.5*theta*exp(-theta*x)}/{0.5*exp(-
lambda*x) + 0.5*exp(-theta*x)}
My S(x) appears to have the right properties of survival functions,
such as tending to 0 when x gets large. Also, S(0)=1. However, I have
a hunch that my h(x) is wrong.
I greatly appreciate your help. |
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