Stats 241 Review Problems for Final Exam
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1. Find the mean and variance of the random variable X whose probability density
function is
(
)
(
)
fxx x x
otherwise
( ) =− − ≤ ≤
3
41 3 1 3
0
2. A random variable X has probability density function
fx
x x
x x
otherwise
( ) =
+ − < <
− ≤ <
1 1 0
1 0 1
0
Find E[X2 - 2X + 2].
3. A random variable has MX(t) = 2
5e t + 1
5e2t + 2
5e3t
a) What is the probability distribution function for X?
b) Expand MX(t) in a power series and find µ and σ2.
4. Suppose that X is a continuous random variable with probability density function:
f(x) = 1
2
e
−
x if -∞ < x < ∞.
Note: |x| = the absolute value of x =
<
≥
0 xifx-
0 xifx
a) Show that X has moment generating function:
MX(t) =
−
−
otherwise
if
t
undefined
1<t<1
1
1
2
b) Use MX(t) to find:
i) µ = E[X]
ii) µ2 = E[X2],
iii) µk = E[Xk] = ‘the kth moment of X.’
5. Suppose the continuous random variables X and Y, have joint probability density
function f(x,y) given below:
f(x,y) =
−≤≤
otherwise
xyforKx
0
10 22
a) Find the value of K.
