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Characteristics of a Binomial Random Variable
1. The experiment consists of n identical trials.
2. There are only two possible outcomes on each trial. Denote one outcome by S (for
Success) and the other by F (for Failure).
3. The probability of S remains the same from trial to trial. Denote it by p=Pr(Success).
4. The trials are independent each other.
Then we define Binomial Random Variable X as the number of Successes in n trials.
Examples
For each of the followings, identify whether the given random variables are BINOMIAL
or not.
1. Recording the genders of 250 newborn babies.
2. A political polling organization calls 1012 people and asks, “Do you approve,
disapprove, or have no opinion of the way the president is handling his job?” The
random variable Y represents the number of people who approve of the way the
president is handling his job.
3. A state lottery randomly chooses six balls numbered from 1 to 40. You choose six
numbers and purchase a lottery ticket. The random variable represents the number of
matches on your ticket to the numbers drawn in the lottery.
Suppose that X=the number of successes among n independent Bernoulli trials with a
success probability p=P(success). Then we denote
X B(n, p)
and
x
P X x nCx p 1 p Also,
E X n p and Var X n p 1 p
4. Suppose X B(5, 0.2). Then calculate
a. List the possible values that X can take.
b. P(X=0)
c. P(X>0)
d. P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4)+P(X=5)
e. Calculate E X and Var X .
n x
, x 0,1,2,. .. , n .
5. Ten percent of American adults are left-handed. A statistics class has 15 students in
attendance.
a. Let X=the number of left-handed students among 15 students in the class.
b. Find the probability that the the number of left-handed students in the class are more
than 5.
c. Find the mean and standard deviation of X.
Example of discrete random variable
6. Suppose that a random variable X has the following probability distribution.
X
1
2
3
4
P(X)
0.15
2k
0.52
k
a. Find the value of k so that the above is a probability distribution.
b. Using the value of k in A, calculate P(X < 3) and P(X > 3).
c. Calculate E(X) and Var(X).