PROBLEM SET 1
0
0.2

1. Suppose that the CDF of a random variable X is given by FX  x   
0.7
1
Find
a)
P  X  1
b)
P  X  1.5
c)
P  1.8  X  1.9
d)
The probability mass function f X  x  of X
x  2
2 x 0
.
0 x2
x2
2. Let 𝑋 be a random variable with probability mass function 𝑓(𝑥) = 𝑘(5 − 𝑥), 𝑥 = 1,2,3,4
a) What must the value of k be in order for f to be a probability mass function?
b) Calculate
i)
The probability that the value of X is odd
ii)
EX 
Var  X 
iii)
c) Find the CDF FX  x  of X .
3. Suppose Z is a random variable with the following probability distribution
𝑥
𝑓(𝑥)
3
0.04
4
0.10
5
0.26
6
0.31
7
0.22
8
0.05
9
0.02
a) Calculate P  7  Z  9 , P  7  Z  9  and P  Z  7 or Z  9 
b) Determine the CDF of Z and sketch its graph.
 
c) Find E  X  , E X 3 and Var  X  .
4. The probability density function f of a continuous random variable X is given by
5

2
k 2 x  x , 0  x 
fX  x  
2

0
elsewhere



where k is a constant.
a) What must be the value of k ?
b) Determine the CDF FX of X and sketch its graph.
c) Calculate P  1  X  1 .
5. The probability density function f of a continuous random variable X is given by 𝑓(𝑥) = 𝑥 − 𝑘𝑥 3 , 0 < 𝑥 < 2
where k is a constant.
d) What must be the value of k ?
e) Determine the CDF FX of X .
f)
Sketch the graphs of FX .
g) Calculate P  X  1.3 , P  1  X  1 .
x2
, 2 x  4

6. Suppose that X is a random variable with probability density function f  x    18
.
0
otherwise
Find
a)
P  X  1
b)
P X2 9
c)
E  X  and Var  X 


2
7. If E  X   1 and Var  X   5 , find E  2  X   and Var  4  3 X   .




3

4 x  x , 0  x  1
8. Find the mean and the median of the probability density function f  x   
.
otherwise

0
 
 
9. Suppose that X is a random variable for which E  X   1, E X 2  2 and E X 3  5 . Find the value of the third
central moment of X .
 1  x
, x  1, 2,3,....

10. Let X be a discrete random variable with probability mass function f  x    2 
.
0
elsewhere

a) Find M X  t 
b) Use part (a) to find Var  X 
11. A random variable X has the MGF given by M X  t   1  t  . Find the mean and variance of X . Identify the
probability distribution of X .
3
 4 
x
4 x
 0.2 0.8 , x  0,1, 2, 3, 4
12. Let X be a discrete random variable with probability mass function f x    x 
0
otherwise

a) Compute the MGF of X
b) Find the MGY of the random variable Y  4  X
13. A random variable X has its MGF given by M t  


1 t
e  2e 4t  2e 8t for    t   ,
5
a) Find  and  2
b) Identify the probability mass function of X
14. Let X be a random variable with mean  and variance  2 , and let M X t  denote the MGF of X for
   t   . Let c be a given positive constant, and let Y be a random variable for which the MGF is
M Y t   e cM X t 1 for    t   . Find the expressions for the mean and variance of Y in terms of the mean
and variance of X ,
15. Let X be a random variable such that P X  0  0 and let   E X  exist. Show that P  X  2   
 
1
.
2
16. If X is a random variable such that E X   3 and E X 2  13 , use Chebyshev’s inequality to determine a lower
bound for the probability P 2  X  8 .
17. From past experience a professor knows the test score of a student taking her final examination is a random
variable with mean 75.
a) Give an upper bound for the probability that a student’s test score will exceed 85.
b) Suppose that the professor knows that the variance of a student’s test score is equal to 25. What can be said
about the probability that a student will score between 65 and 85?
5
, 𝑥>0
18. Let X be a random variable with probability density function 𝑓(𝑥) = {𝑥 6
. What bound does
0
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Chebyshev’s inequality give to the probability 𝑃(𝑋 > 2.5)?