Random Variables Notes
A discrete random variable X has a countable number of possible values. The probability distribution of X
lists the values and their probabilities.
Value of X
x1
x2
x3
…
xn
Probability
p1
p2
p3
…
pn
Where 0  pi  1 for all I and p1  p2  p3    pn  1 .
On a 5–item multiple choice exam there are choices A-D on each item. A student who blindly guesses on
each item may get anywhere from 0 to 5 correct. Let X = the number correct. The probability distribution
of X is…
Value of X
Probability
0
.23730
1
.39551
2
.26367
3
.08789
4
.01465
(Verify these probabilities using the binompdf function on your calculator.)
Verify that this is a legitimate probability distribution for this random variable.
Find the probability that X is 2 or 3.
Find the probability that X is greater than 3.
Find the probability that the student gets at least 1 correct.
5
.00098
Means and Variances of Random Variables:
Suppose the probability distribution of X is
Value of X
x1
x2
x3
…
xn
Probability
p1
p2
p3
…
pn
The mean of X is  X   xi  pi
and the variance of X is  X2   ( xi   X ) 2  pi
and the standard deviation of X is  X   X2 
 (x
i
  X ) 2  pi
On a 5–item multiple choice exam there are choices A-D on each item. A student who blindly guesses on
each item may get anywhere from 0 to 5 correct. Let X = the number correct. The probability distribution
of X is…
Value of X
Probability
0
.23730
1
.39551
2
.26367
3
.08789
4
.01465
5
.00098
Find  X and  X .
x
p
0
.23730
1
.39551
2
.26367
3
.08789
4
.01465
5
.00098
xi  p i
X 
( xi   x ) 2  p i
 X2 
X 
Rules for means and variances:
Suppose X has the distribution…
X
Prob.
2
0.25
3
0.75
And suppose Y has the distribution…
Y
Prob.
4
0.30
5
0.20
6
0.50
Verify each of the following:
 X  2.75
 X2  0.187
Y  5.20
 Y2  0.760
Find the mean and variance of 3X (mean = 8.25 var = 1.687)
3X
Prob.
6
0.25
9
0.75
Find the mean and variance of 5Y (mean = 26 var = 19.0)
5Y
Prob.
20
0.30
25
0.20
30
0.50
Find the mean and variance of 2 + 7X (mean = 21.25 var = 9.187)
2+7X
Prob.
16
0.25
23
0.75
In general, if X is a random variable then
 a bX 
 a2 bX 
Do these rules hold for finding the mean and variance of 3 + 2Y? Try it.
Suppose X has the distribution…
X
Prob.
2
0.25
3
0.75
And suppose Y has the distribution…
Y
Prob.
4
0.30
5
0.20
6
0.50
Find each of the following
X 
 X2 
Y 
 Y2 
If X and Y are independent and added then the sum X+Y takes the following values…
Find the probabilities
X+Y
6
7
8
9
Prob
0.075
0.275
0.275
0.375
Find the mean and variance of X + Y.
 X Y 
 X2 Y 
In general, if X and Y are independent random variables, then
 X Y 
 X2 Y 
Does the same hold for subtraction? Try it.
X-Y
-1
-2
-3
-4
Prob
.225
.225
.425
.125
In statistics…
3X  X + X + X
Consider a game of chance in which the player draws numbers from a hat. The numbers in the hat are
numbered 1, 2, and 4 with the following proportions:
X
P(X)
1
0.5
2
0.4
4
0.1
Find each of the following
 X  _________
 X  _________
 X2  _________
Case 1: Draw 1 number from the hat and triple the value so Y = 3X
Y = 3X
P(Y)
3
0.5
6
0.4
12
0.1
Find each of the following
Y  _________
 Y  _________
 Y2  _________
Case 2: Draw 3 numbers (with replacement) and add the values so Z =
Z
P(Z)
3
0.125
4
0.300
5
0.240
6
0.139
7
0.120
8
0.048
X1  X 2  X 3
9
0.015
10
0.012
Find each of the following
 Z  _________
 Z  _________
 Z2  _________
 nX  _________
2
 nX
 _________
X
 X2
In general…
 nX  _________
and
X
1  X 2  X n
 _________
1  X 2  X n
 _________
1  X 2  X n
 _________
12
0.001
Suppose X is normally distributed with  X  50 and  X  8 and Y is normally distributed with Y  30
and  Y  6 .
Describe the distribution of X + Y.
What is the probability that X + Y > 95?
Describe the distribution of X – Y.
What is the probability that X > Y?
Suppose the time (A) in hours required to assemble a computer is normally distributed with  A  6 and
 A  1.2 and the time (B) in hours required to box it and get it on a truck is also normally distributed with
 B  2 and  B  0.5 . Assume that the time required to complete each task is independent of the other.
What is the probability that the entire process will take more than 10 hours?
What is the probability that it will take less than 15 hours to assemble 3 computers?
What is the probability that this crew could assemble a computer quicker than it could box and load 2
computers.
If A is the time required to assemble 1 computer, then the average time required to assemble 3 computers
1
1
1
1
is M  ( A1  A2  A3 )  A1  A2  A3 .
3
3
3
3
Find  M and  M .
What is the probability that M > 6.25?