Ch. 3: pt 2 - Mathematical Sciences Home Pages

Chapter 3: DISCRETE
RANDOM VARIABLES AND
PROBABILITY DISTRIBUTIONS
Part 2: Mean and Variance of a
Discrete Random Variable
Section 3-4
In a random experiment, there are a variety of
possible outcomes.
What outcome do we expect to see?
Some outcomes may be more likely than others.
Note: an expectation or expected value
has a specific mathematical meaning...
1

• Mean of a Discrete Random Variable
The mean or expected value of a discrete random
variable X, denoted as µ or E(X), is
µ = E(X) = ∑ x
x · f(x) = ∑ x
x · P (X = x)
Example: Consider the random variable X
and associated probability mass function defined
by P (X = 0) = 0.20, P (X = 1) = 0.30, and
P (X = 2) = 0.50.
The expected value of X or E(X) by the definition
above is
E(X) = 0 · P (X = 0) + 1 · P (X = 1) + 2 · P (X = 2)
= 0 · 0.20 + 1 · 0.30 + 2 · 0.50 = 1.30 ∗
∗ Note that eventhough X can only take on a value of 0, 1, or 2,
the expected value can fall anywhere on the real number line.
2

E(X) = 1.30
Though the possible values for X are 0, 1, and
2, the expected value is closer to 2 because 2 is
more ‘heavily weighted’, or more likely to occur.
The mean of a discrete random variable X is
a weighted average of the possible values of X,
with weights equal to the probabilities.
A probability distribution can be viewed as a
loading with a mean equal to the balance point
(shown as dark triangles). Parts (a) and (b)
above illustrate equal means from very different
loadings (or distributions).
3

In statistics, the mean is a measure of center
for a distribution.
• Example: Cylinders in the engine of a car
Let X be the number of cylinders in the engine
of the next car to be tuned up at a certain
facility. Based on past data, the probability
mass function for X is the following:
x 4 6 8
f(x) 0.55 0.25 0.20
The cost, h, of a tune-up is a function of X.
Specifically, we let the cost be represented as
the function h(X), and
h(X) = 20 + 3X + 0.5X 2
{ cars with more cylinders cost more to tune-up }
Find the expected cost of the next tune-up,
or E[h(X)].
4

Find E[h(X)].
5

• Expected Value of a Function of a
Discrete Random Variable
- If X is a discrete random variable with
probability mass function f(x),
E[(h(X)] = ∑ x
= ∑ x
h(x) · f(x)
h(x) · P (X = x)
You can use this formula to find the expected
value of any function of a discrete random
variable X.
In statistics, a very special function of X for
which we take the expected value is (X −µ) 2
where µ was defined earlier as E[X].
We use E[(X − µ) 2 ] to quantify the spread
in a distribution. And this value is called the
variance of a distribution...
6

As mentioned earlier, the mean is measure of
center for a distribution.
But what about the spread?
The two probability mass functions shown above
(viewed as a loading on a beam) have the same
mean, but not the same spread.
There are a variety of ways to quantify spread
or dispersion. Consider a simple one..
Range: x max − x min
This one is minimally informative.
7

Better yet, we could relate the values in a distribution
to a measure of center (then all values
contribute to the measure of spread).
For example...
Expected absolute distance from the mean:
Expected value of |x i − EX|
{ On average, how far from center are the values?}
But probably the most often used variability
measure is the variance of a distribution.
8

• Variance
The variance of X, denoted as σ 2 or V (X),
is
σ 2 = V (X) = E(X − µ) 2
It is the expected squared distance of X from
the mean (or expected value) of the random
variable.
Since X is a discrete random variable,
E(X−µ) 2 = ∑ x
(x−µ) 2 f(x) = ∑ x
x 2 f(x)−µ 2
where f(x) is the probability mass function
for X.
To use either of the summation formulas you
must first compute EX = µ.
9

• Standard Deviation
Variance is in units 2 , so we often work with
standard deviation instead which is in the
original units of the data.
The standard deviation of X is denoted as
σ where σ = √ σ 2 = √ V (X).
The standard deviation of X is just the square
root of the variance of X.
∗ Though the ‘average absolute distance from center’ is in
the origin units and may be more easily interpreted, it
turns out that the squared distance function is nicer to
deal with than the absolute value function mathematically.
10

Remember, variance is a measure of spread or
dispersion.
• Example: Flaws in a piece of wire.
Let X represent the number of flaws in a randomly
chosen piece of wire.
x f(x)
0 0.48
1 0.39
2 0.12
3 0.01
Compute the variance and standard deviation
of X.
ANS:
11

e.g. (cont.)
12

• Example: Coin toss game.
You and a friend play a game where you each
toss a balanced coin. If the upper faces on
the coins are both tails, you win $1; if the
faces are both heads, you win $2; if the coins
do not match (one shows a head the other a
tail), you lose $1.
Let X represent the amount won.
1. What is the probability distribution for
X?
13

2. Find the mean and variance of X.
14

e.g. (cont.)
15