Anyone can help me with my math homework? thanks a lot!

MATH 464
HOMEWORK 8
SPRING 2016

The following assignment is to be turned in on
Thursday, April 7, 2016.
1. Let X be an exponential random variable with parameter ? &gt; 0.
a) Let t ? 0 and calculate P (X ? t).
b) Let s, t ? 0 and calculate P (X ? s + t|X ? s). (You can compare your
answer to this question with your answer to problem #5 on homework #5.)
2. Let X be a normally distributed random variable with real parameters
? and ? &gt; 0. Find the mean and variance of X. Hint: It may be useful to
remember that
?
?
2
e?x dx = ? .
??

3. The gamma function is de?ned by
?

?(w) =

xw?1 e?x dx

0

for all w &gt; 0. In terms of this function, a continuous random variable X
(with parameters w &gt; 0 and ? &gt; 0) is de?ned by setting
fX (x) =

?w w?1 ??x
e
?(w) x

0

if x &gt; 0,
otherwise.

and declaring that X has probability density function fX (x). (fX is called
the gamma distribution with parameters w &gt; 0 and ? &gt; 0.)
a) Show that X is a continuous random variable by showing that
fX (t) dt = 1
R

for all values of w &gt; 0 and ? &gt; 0.
b) Show that for any w &gt; 1,
?(w) = (w ? 1)?(w ? 1)
Use your result to calculate ?(n) for any integer n ? 2.
c) Compute the mean and variance of this random variable X.
1

2

SPRING 2016

4. Let X be uniformly distributed on [0, 1]. Find the cumulative distribution
function (i.e. the cdf) for the random variable
3X
Y =
1?X

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