Probability Distribution
Probability Distribution
Probability Distribution
Distributions
Probability
Distributions
Random Variables
A random variable x represents a numerical value
associated with each outcome of a probability distribution.
In Words In Symbols
1. The probability of each value of 0 P (x) 1
the discrete random variable is
between 0 and 1, inclusive.
Guidelines
Let x be a discrete random variable with possible
outcomes x1, x2, … , xn.
1. Make a frequency distribution for the possible
outcomes.
2. Find the sum of the frequencies.
3. Find the probability of each possible outcome by
dividing its frequency by the sum of the frequencies.
4. Check that each probability is between 0 and 1 and
that the sum is 1.
Example:
The spinner below is divided into two sections. The
probability of landing on the 1 is 0.25. The probability of
landing on the 2 is 0.75. Let x be the number the spinner
lands on. Construct a probability distribution for the
random variable x.
1
x P ( x)
1 0.25 Each probability is
2 between 0 and 1.
2 0.75
Example:
The spinner below is spun two times. The probability of
landing on the 1 is 0.25. The probability of landing on the 2
is 0.75. Let x be the sum of the two spins. Construct a
probability distribution for the random variable x.
Continued.
Larson & Farber, Elementary Statistics: Picturing the World, 3e 8
Constructing a Discrete Probability Distribution
Example continued:
Example continued:
Sum of
P ( x)
spins, x
2 0.0625 Each probability is between
3 0.375 0 and 1, and the sum of the
4 0.5625 probabilities is 1.
Example:
Graph the following probability distribution using a histogram.
Sum of P(x)
P ( x) Sum of Two Spins
spins, x 0.6
2 0.0625 0.5
3 0.375
Probability
4 0.5625 0.4
0.3
0.2
0.1
0 x
2 3 4
Sum
Larson & Farber, Elementary Statistics: Picturing the World, 3e 11
Mean
The mean of a discrete random variable is given by
μ = ΣxP(x).
Each value of x is multiplied by its corresponding
probability and the products are added.
Example:
Find the mean of the probability distribution for the sum of
the two spins.
x P ( x) xP (x)
2 0.0625 2(0.0625) = 0.125 ΣxP(x) = 3.5
3 0.375 3(0.375) = 1.125 The mean for the
4 0.5625 4(0.5625) = 2.25 two spins is 3.5.
Larson & Farber, Elementary Statistics: Picturing the World, 3e 12
Variance
The variance of a discrete random variable is given by
2 = Σ(x – μ)2P (x).
Example:
Find the variance of the probability distribution for the sum
of the two spins. The mean is 3.5.
Example:
At a raffle, 500 tickets are sold for $1 each for two prizes of
$100 and $50. What is the expected value of your gain?
Gain, x P ( x)
E(x) = ΣxP(x).
1
$99 500 1 1 498
$99 $49 ($1)
1 500 500 500
$49 500
$0.70
–$1 498
500
Because the expected value is
Winning negative, you can expect to lose
no prize
$0.70 for each ticket you buy.
Larson & Farber, Elementary Statistics: Picturing the World, 3e 16
§ 4.2
Binomial
Distributions
Binomial Experiments
A binomial experiment is a probability experiment that
satisfies the following conditions.
1. The experiment is repeated for a fixed number of
trials, where each trial is independent of other trials.
2. There are only two possible outcomes of interest for
each trial. The outcomes can be classified as a success
(S) or as a failure (F).
3. The probability of a success P (S) is the same for each
trial.
4. The random variable x counts the number of
successful trials.
Symbol Description
n The number of times a trial is repeated.
0.1
0 x
0 1 2 3 4
Number of red chips
Larson & Farber, Elementary Statistics: Picturing the World, 3e 25
Mean, Variance and Standard Deviation
Population Parameters of a Binomial Distribution
Mean: μ np
Variance: σ 2 npq
Standard deviation: σ npq
Example:
One out of 5 students at a local college say that they skip breakfast in
the morning. Find the mean, variance and standard deviation if 10
students are randomly selected.
n 10 μ np σ 2 npq σ npq
p 1 0.2 10(0.2) (10)(0.2)(0.8) 1.6
5
q 0.8 2 1.6 1.3