Download Math 511 Problem Set 10 Solutions

Math 511 – Problems 10
1. Suppose that X is a random variable that has the probability function
⎧0.3 if k = 8
⎪
P(X = k) = p(k) = ⎨0.2 if k = 10
⎪0.5 if k = 6
⎩
What is the moment generating function for X?
Solution: M X (t) = 0.3e8t + 0.2e10t + 0.5e 6t .
2. Suppose that Y is a random variable with moment generating function H(t). Suppose
further that X is a random variable with moment generating function M(t) given by
1
M(t) = (2e 3 t + 1)H (t) . Given that the mean of Y is 10 and the variance of Y is 12,
3
then determine the mean and variance of X.
Solution:
Since the mean of Y is 10, H '(0) = 10 . Since the variance of Y is 12,
12 = E(Y 2 ) − E(Y )2 = E(Y 2 ) − 100 ⇒ E (Y 2 ) = 112 . Hence, H ''(0) = 112 .
d ⎡1
1
⎤
M '(t) = ⎢ (2e 3t + 1)H (t)⎥ = 2e 3t H (t) + (2e 3t + 1)H '(t)
dt ⎣ 3
3
⎦
d⎡
1
1
⎤
M '' (t) = ⎢ 2e 3t H (t) + (2e 3t + 1)H '(t)⎥ = 6e 3t H (t) + 4e 3t H ' (t) + (2e 3t + 1)H '' (t)
dt ⎣
3
3
⎦
'
So, E(X ) = M (0) = 2H (0) + H ' (0) = 2 + 10 = 12 .
E(X ) = M '(0) = 2H (0) + H ' (0) = 2 + 10 = 12
E(X 2 ) = M '' (0) = 6H (0) + 4H ' (0) + H '' (0) = 6 + 40 + 112 = 158 .
So, Var(X) = E(X 2 ) − E(X)2 = 158 −144 = 14 .
3. Suppose that the Moment generating function for X is M (t) =
et
.
3 − 2et
Then determine µ and σ 2 for X.
Solution:
While we could do this by taking derivatives, it is quicker to notice that
1 et
et
3
M (t) =
=
. And so X must be a geometric random variable with
t
3 − 2e 1 − 2 3 et
2
1
1
2
probability p = of success. So, µ = 1 = 3 , and σ = 1 3 = 6 .
3
9
3
4. Suppose that the moment generating function of the random variable X is
⎛ 1 + 2et ⎞ ⎛ 1 + 3et ⎞
.
M (t) = ⎜
⎝ 3 ⎟⎠ ⎜⎝ 4 ⎟⎠
(a). What is the probability P(X = 1) ?
(b). What is the probability P(X = 2) ?
(c). What is the probability P(X = 3) ?
Hint: Expand the product defining M(t).
⎛ 1 + 2et ⎞ ⎛ 1 + 3et ⎞ 1 + 5et + 6e2t
Solution: M (t) = ⎜
.
=
12
⎝ 3 ⎟⎠ ⎜⎝ 4 ⎟⎠
5
1
So, (a). P(X = 1) =
,
(b). P(X = 2) = ,
12
2
(c). P(X = 3) = 0.
5. Suppose that the moment generating function of the random variable X is
10
⎛ 1 + 3et ⎞
. What is the mean and variance of X?
M (t) = ⎜
⎝ 4 ⎟⎠
Solution: this is the moment generating function for a binomial random variable
15
witb n = 10 and p = 0.75. So, µ = 10 × 0.75 = 7.5 and σ 2 = 10 × 0.75 × 0.25 =
.
8
In Your Text:
3.151, 3.153, 3.155, 3.158, 3.159
Read Section 3.9