Page 174 - DMTH404_STATISTICS

Statistics

Notes 12.8 Review Questions

1. Let X be a discrete random variable having a Bernoulli distribution. Its support R is
X
R = <0, 1>
X
and its probability mass function p (x) is
X

 p if x  1


p (x)   1 p if x  0
X


 0 if x R X
where p  (0, 1) is a constant. Derive the moment generating function of X, if it exists.
2. Let X be a discrete random variable having a Bernoulli distribution. Its support R is
X
R = <0, 1>
X
and its probability mass function p (x) is
X
 p if x  0


p (x)   1 p if x  1
X
 0 if x R

 X
where p  (0, 1) is a constant. Derive the moment generating function of X, if it exists.
3. Let X be a random variable with moment generatinf function
1
M (t) = 1 exp(t) 
X 3
Derive the variance of X.
4. Let X be a random variable with moment generatinf function

4
M (t) = 1 exp(t) 
X 3
Derive the variance of X.
5. A random variable X is said to have a Chi-square distribution with n degrees of freedom
1
if its moment generating function is defined for any t  and it is equal to:
2
M (t) = (1 – 3t) -n/2
X
Answers: Self Assessment

1. (d) 2. (a) 3. (b) 4. (c)

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