Question 6, 7, 8
6. Let the random variable X have the p.d.f.
f ( x ) =
8
2
2
1
?
x
e
?
, 0 < x < ?, zero elsewhere.
Find the mean and the variance of X.
Hint: Compute E ( X ) directly and E ( X
2
) by comparing the integral with the integral
representing the variance of a random variable that is N ( 0, 4 ).
7 – 8. (i) Give the name of the distribution of X (if it has a name), (ii) find the values
of µ and ?
2
, and (iii) calculate P ( 1 ? X ? 2 ) when the moment-generating
function of X is given by
7. a) M X ( t ) =
1 2.5 t
1
?
, t < 0.4.
b) M X ( t ) =
3
1 25.0
1
?
?
?
?
?
?
? t
, t < 4.
8. c) M X ( t ) = e
3 t + 2 t
2
.
d) M X ( t ) =
t
t
e
5
1
5
?
, t ? 0, M X ( 0 ) = 1.
