Download Mean of a discrete random variable

Chapter 7 notes
Variable—A quantity whose value changes.
Discrete variable X—has a countable number of possible values. Discrete means
count 1, 2, 3, etc. .
Examples: number of students in the class
number of green m & m’s in the package
number of heads when flipping a coin ten times
Continuous variable—a variable whose value is obtained by measuring.
Examples: height of students in the class
arm span of students in the class
time it takes to travel to school
Random Variable—a variable whose value is a numerical outcome of a random
phenomenon. It associates a number with each outcome in the sample space.
A random variable is denoted with a capital letter
A random variable can be discrete or continuous.
Probability Distribution of X this lists the values of X and their probabilities.
Value of X: x1
x2
x3
x4
Probability: p1
p2
p3
p4
The probabilities pi must satisfy two requirements
1. Every probability pi is a number between 0 and 1.
2. p1 + p2 + p3 + ….+pk = 1
Example: Let X represent the sum of two dice
Then the probability distribution of X is as follows:
X
2
3
4
5
6
7
8
9
10
11
12
P(X)
1
36
2
36
3
36
4
36
5
36
6
36
5
36
4
36
3
36
2
36
1
36
To graph the probability distribution of a discrete random variable, construct a
probability histogram.
Probability Distribution of X
Probability
0.20
0.15
0.10
0.05
0.00
2
3
4
5
6
7
Outcome
8
9
10
11
12
A mail order computer business has six telephone lines. Let x denote the number
of lines in use at a specified time. The probability distribution is in the following
table.
x
p(x)
0
.10
1
.15
2
.20
3
.25
4
.20
5
.06
6
.04
Calculate the probability of each of the following events.
a) P(at most three lines are in use)
b) P(fewer than three lines are in use)
c) P( at least three lines are in use)
d) P(between two and four lines, inclusive, are in use)
e) P(between two and four lines, inclusive, are not in use)
Mean of a discrete random variable—Is the weighted average. Each value of x
is weighted by its probability
To find the mean of X, multiply each value of X by its probability, then add
all the products.
 X  x1 p1  x2 p2    xk pk
  xi pi
The mean of a random variable X is called the expected value of X.
Expected value—of a random variable that has a numerical outcome is found by
multiplying each outcome by its probability and then summing all possible
outcomes. It is an average of the possible outcomes, but not the ordinary average
that you are use to where everything is equal. The expected value represents the
“long-run average” if we repeat the actual event many times.
Example: What is the mean of motor vehicles in American households as of 1997.
We are ignoring the very few that have more than 5 vehicles.
Number of 0
vehicles
Proportion 0.04
of
households
1
2
3
4
5
0.25
0.45
0.18
0.06
0.02
The expected value is (0)(0.04) + (1)(0.25) + (2)(0.45) + (3)(0.18) + (4)(0.06) +
(5)(0.02) = 2.03
Law of large numbers—if an experiment is repeated many times, then the relative
frequency with which an event occurs equals the probability of the event.
The Variance of a Discrete Random Variable:
If X is a discrete random variable with mean
 , then the variance of X is
 2   x1   X  p1   x2   X  p2     xk   X  pk
2
2
X
   xi   X  pi
2
The standard deviation
 X  is the square root of the variance.
2
Bernoulli random variable—any random variable whose only possible values are
0 or 1. For example, when tossing a coin, with X being heads, you either have
heads (value of 1) or not heads (value of 0).