suppose life time of certain make of T.V. tube is exponentially distributed with mean life 1600 hrs what is the probability that 1) the tube will work upto 2400 hrs. 2) the tube will survive after 1000hrs?
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STAT 400
UIUC Answers for 3.2 (Part 1) Stepanov
Dalpiaz
Uniform Distribution over an interval [ a , b ]:
For Uniform distribution,
P( c ≤ X ≤ d ) =
d c
b a
−
− , a ≤ c ≤ d ≤ b.
E(X) =
a b
2
+ , Var(X) = 12
b a 2 ( − ) .
0.25. Let X be a random variable distributed uniformly over the interval [ a, b ].
Find the moment-generating function of X.
MX (t) = E( e t X ) = ( ) ∫
∞
∞
⋅
-
f x dx t x e = ∫ −
⋅
b
a
t x dx
b a
e
1
=
a
b
b a t
t x 1 e
−
⋅ = ( )
t b a
t b t a e e
−
− , t ≠ 0.
MX ( 0 ) = 1.
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