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Chapter 4 Discrete Probability Distributions
Random Variable (RV): A random variable
has a single numerical value for each outcome of
an experiment. That is, a random variable is a
function that associates a real # with each
outcome of the experiment.
Discrete Random Variable (DRV): A random
variable that assumes only a finite or countably
infinite # of values.
Continuous Random Variable (CRV): A
random variable that has infinitely many values,
and those values can be associated with points
on a continuous interval in such a way that there
are no gaps or interruptions.
Example 1: Determine whether the random
variable is discrete or continuous:
(a) # of defective vending machines at
VSU;
(b) The amount of time required to
complete your homework each day for
your math class;
(c) Heights of male students enrolled at
VSU in Fall 2004;
(d) The number of students in your math
class at VSU;
(e) Distance required to stop a car travelling
at 70 mph;
(f) # of chess games you will have to play
before winning a game of chess.
Probability Distribution: A listing of the
possible values and corresponding probabilities
of a discrete random variable.
Example 2: Toss two coins. Let X = “# of
heads.” List the values of X and the associated
probabilities.
Soln. S = {HH, HT, TH, TT}
X f(x)
Outcome(s)
0
¼
TT
1
2/4
HT, TH
2
¼
HH
Note: The function f above is called probability
(density) function for X.
Discrete Probability Distributions Let X be a
discrete random variable. For the function f(x) to
be a probability (density) function for X, the
following two conditions must be satisfied:
1. f(x)  0 for each x
2.  f (x) = 1
Example 3: Given the probability distribution of
a discrete random variable Y, find
(a) P(Y= 2) (b) P( 2  Y  4) (c) P(Y < 5)
y
0
1
f(y)
0.1
0.2
2
3
4
5
?
0.3
0.25
0.1
Mean and Variance of a Discrete RV X
Expected Value (Mean):
 = E(X) =  ( xf ( x))
Note: The expected value of X is a weighted
average of the possible outcomes, with the
probability weights reflecting how likely each
outcome is. The expected value of X is also
called the mean of X.
Variance:
2
= VAR(X) = (x  )
2
f ( x)
An easier formula for calculating Variance
2
= VAR(X) = (x
2
f ( x))   2
Standard Deviation

= S.D.(X) =
Variance
Example 4: Reference Example 2, find E(X)
and  .
Soln.
X f(x)
0 ¼
xf(x) x f(x)
0
0
1
2/4
2/4
2/4
2
¼
2/4
4/4
2

1
6/4
 = E(X) =  ( xf ( x)) = 1
Note: The expected value of X should be
interpreted as the long run average value of X.
2
= VAR(X) = (x
 = S.D.(X) =
0.5
2
f ( x))   2
= 0.71
= 6/4 - (1) = 0.5
2
Example 5: Roll two dice. Let X = “sum of the
two numbers.” Find the probability function of
X. Then find E(X) and S.D.(X).
Soln.
x f(x)
xf(x) x f(x)
2 1/36 2/36 4/36
3 2/36 6/36 18/36
4 3/36 12/36 48/36
5 4/36 20/36 100/36
6 5/36 30/36 180/36
7 6/36 42/36 294/36
8 5/36 40/36 320/36
9 4/36 36/36 324/36
10 3/36 30/36 300/36
11 2/36 22/36 242/36
12 1/36 12/36 144/36

252/36 1974/36
2
 = E(X) =  ( xf ( x)) = 252/36 = 7
2
= VAR(X) = (x f (x))   = 1974/36 – 49
= 54.83 – 49 = 5.83
2
2
 = S.D.(X) =
5.83
= 2.42
HW: 1-18(all), 21-27 (odd) pp. 169-170
4.2 The Binomial Distributions
There are many uncertain situations that have
the same characteristics as the classic coinflipping example. We shall study a general
model that is similar to working with coinflipping example.
Properties of a Binonial Experiment
1. The experiment consists of a sequence of n
identical trials, that is, repeat a process n
times.
2. There are two outcomes possible on each
trial -- success or failure.
3. p = P[success] remains constant from trial to
trial.
4. The trials are independent. (That is, the
outcome on one trial has no effect on
outcome of another trial).
In a binomial experiment, our interest is in the
number of successes occurring in n-trials. Let X
= # of successes. Then X is a discrete rv and can
take on the values of 0, 1, 2, …, n. We shall
study the probability distribution of X.
Example 6: See Example on page 174
Factorial of n: Let n be non-negative integer.
We define the factorial of n (n!) as follows:
0! = 1
1! = 1
2! = 2*1 = 2
3! = 3*2*1 = 6
4! = 4*3*2*1 = 24
5! = 5*4*3*2*1 = 120
.
.
.
In general, n! = 1n(ifn n 1)(0n  2)...1 if n  1

Notation: Let n be a non-negative integer and let
0  x  n. Then  nx  denotes the number of ways
 
we can choose x things from a set of n things.
A Formula for
n
 
 x
Note:
=
n
 
 x
n
 
 x
n!
x!( n  x )!
= the number of experimental
outcomes with x-successes given n trial of a
binomial experiment.
Binomial Probability Formula: (Ref. Page 176)
Let us consider a binomial experiment with n
trials such that p = p[success]. Let X = “number
of successes.” Let 0  x  n. Then
P(x-successes) = f(x) =
n
 
 x
p (1-p)
x
n x
Example 7: Example 2 on page 176
Example 8: Example 6 on page 180
Mean and S.D. of a Binomial RV X:
 = E(X) = np,  =
np(1  p)
Example 9: Example 8 on page 182
HW: 1-4 (all), 5-13(odd) pp. 183-185 Also for 9,
11, and 13, compute the expected value of the
random variable.