= a
2
E [X
2
] by Corollary 4.1
= a
2
VarX. by the definition of variance
4. Definition of standard deviation
Definition 4.8
The square root of VarX, Var
X , is called the standard deviation of X, and is denoted as SDX, i.e.,
SDX = Var
X .
G. The Bernoulli and Binomial Random Variables
1. Assumptions for the following discussions
Given a trial with an outcome of success or failure, define a random variable X to be X
= 1 if the outcome = a success; and X
= 0 if the outcome = a failure.
And assume the following pmf for random variable X: p
0 = P{X p
; and p
1 = P{X = 1} = p 4.1
where p is the probability of success in a trial.
2. Definitions of Bernoulli and binomial random variables
Definition 4.9 ---
A random variable X is said to be a Bernoulli random variable if its pmf is described by 4.1 above for some p such that 0 p 1.
Definition 4.10 ---
If X represents the number of successes in n independent trials with p as the p
as that of failure in a trial, then X is called a binomial random variable with parameters n, p.
A comment: a Bernoulli random variable is just a binomial random variable with parameters 1, p
3. The pmf of a binomial random variable
Fact 4.6
The pmf pi for a binomial random variable X with parameters n, p is:
p i = P{X = i}
= P{successes in n trials = i} = Cn, ip
i
p
n -i
,
i =1, 2, ...
4.2 Why?
Think about it by yourself using a similar reasoning used in Example 3.11.
Example 4.13 “wheel of fortune”
A game called “wheel of fortune” often played in casinos goes likeμ bet a number N within 1 through 6, and then roll 3 dies; if N appears i times, i = 1, 2, 3, then the player win
i units; otherwise, the player loses one unit. Is this game fair?
Solution:
A trial = a roll of a die here.
Success in a trial = N appears in the rolling result.
P {N appears in a trial} = 16.
Let X = units won by the player “” means “lose”, and “+” means “win”.
Let Y = times that N appears in the 3 rollings.
Then, Y is a binomial random variable with parameters 3, 16 by definition.
p 1 = P{X
= P{losing one unit} = P{N does not appear in the 3 rollings}
= P{Y = 0} = C3, 016
56
3
by Fact 4.6 = 125216.
p +1 = P{X = +1}
= P{winning one unit} = P{N appears once in the 3 rollings}
= P{Y = 1} = C3, 116
1
56
2
by Fact 4.6 = 75216.
Similarly, p
+2 = P{X = +2} = P{Y = 2}
= C3, 216
2
56
1
by Fact 4.6 = 15216.
p +3 = P{X = +3}
= P{Y = 3} = C3, 316
3
56 by Fact 4.6
= 1216.
To determine if the game is fair, we may compute E[X] as we did in Example 4.8 to see if its value is zero:
E [X] =
: 0 i
x p x
xp x
by the definition of mean =
: 0
{ }
i x p x
xP X x
by the definition of pmf =
125216 + 175216 + 215216 + 31216
This result means that in the long run, the player loses 17 units per every 216 games,
So the game is unfair
4. Properties of binomial random variables
