Download Understanding Random Variables & Binomial Experiments in Discrete Probability and more Lecture notes Statistics in PDF only on Docsity!
Discrete Probability
Distributions
Chapter 4
Probability
Distributions
Larson & Farber, Elementary Statistics: Picturing the World, 3e 3
Random Variables
A random variable x represents a numerical value associated with each outcome of a probability distribution.
A random variable is discrete if it has a finite or countable number of possible outcomes that can be listed.
x 0 2 4 6 8 10
A random variable is continuous if it has an uncountable number or possible outcomes, represented by the intervals on a number line.
x 0 2 4 6 8 10
Larson & Farber, Elementary Statistics: Picturing the World, 3e 4
Random Variables
Example:
Decide if the random variable x is discrete or continuous.
a.) The distance your car travels on a tank of gas
b.) The number of students in a statistics class
The distance your car travels is a continuous random variable because it is a measurement that cannot be counted. (All measurements are continuous random variables.)
The number of students is a discrete random variable because it can be counted.
Larson & Farber, Elementary Statistics: Picturing the World, 3e 5
Discrete Probability Distributions
A discrete probability distribution lists each possible value the random variable can assume, together with its probability. A probability distribution must satisfy the following conditions.
In Words In Symbols
- The probability of each value of the discrete random variable is between 0 and 1, inclusive.
0 ≤ P (x) ≤ 1
- The sum of all the probabilities is 1. ΣP (x) = 1
Larson & Farber, Elementary Statistics: Picturing the World, 3e 6
Constructing a Discrete Probability Distribution
Guidelines Let x be a discrete random variable with possible outcomes x 1 , x 2 , … , xn.
- Make a frequency distribution for the possible outcomes.
- Find the sum of the frequencies.
- Find the probability of each possible outcome by dividing its frequency by the sum of the frequencies.
- Check that each probability is between 0 and 1 and that the sum is 1.
Larson & Farber, Elementary Statistics: Picturing the World, 3e 10
Constructing a Discrete Probability Distribution
Example continued:
(^1) P (sum of 4) = 0.75 × 0.75 = 0.
Spin a 2 on the “and” first spin.
Spin a 2 on the second spin.
3 0. 4
2 0.
P (x) Sum of spins, x
Each probability is between 0 and 1, and the sum of the probabilities is 1.
Larson & Farber, Elementary Statistics: Picturing the World, 3e 11
Graphing a Discrete Probability Distribution
Example:
Graph the following probability distribution using a histogram.
3 0. 4 0.
2 0.
P (x) Sum of spins, x
Sum of Two Spins
0
x
Probability
2 3 4 Sum
P(x)
Larson & Farber, Elementary Statistics: Picturing the World, 3e 12
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.
2 0. 3 0. 4 0.
x P (x)
Example: Find the mean of the probability distribution for the sum of the two spins.
2(0.0625) = 0. 3(0.375) = 1. 4(0.5625) = 2.
xP (x) ΣxP(x) = 3. The mean for the two spins is 3.5.
Larson & Farber, Elementary Statistics: Picturing the World, 3e 13
Variance
The variance of a discrete random variable is given by
σ^2 = Σ(x – μ)^2 P (x).
2 0. 3 0. 4 0.
x P (x)
Example:
Find the variance of the probability distribution for the sum of the two spins. The mean is 3.5.
–1. –0.
x – μ
(x – μ)^2 ≈ 0. ≈ 0. ≈ 0.
P (x)(x – μ)^2 ΣP(x)(x^ – 2) 2
The variance for the two spins is approximately 0.
Larson & Farber, Elementary Statistics: Picturing the World, 3e 14
Standard Deviation
2 0. 3 0. 4 0.
x P (x)
The standard deviation of a discrete random variable is given by
Example:
Find the standard deviation of the probability distribution for the sum of the two spins. The variance is 0.376.
–1. –0.
x – μ
σ = σ^2.
(x – μ)^2
P (x)(x – μ)^2
Most of the sums differ from the mean by no more than 0. points.
σ =σ^2 = 0.376 ≈0.
Larson & Farber, Elementary Statistics: Picturing the World, 3e 15
Expected Value
The expected value of a discrete random variable is equal to the mean of the random variable.
Expected Value = E(x) = μ = ΣxP(x).
Example:
At a raffle, 500 tickets are sold for $1 each for two prizes of $ and $50. What is the expected value of your gain?
Your gain for the $100 prize is $100 – $1 = $99.
Your gain for the $50 prize is $50 – $1 = $49.
Write a probability distribution for the possible gains (or outcomes). Continued.
Larson & Farber, Elementary Statistics: Picturing the World, 3e 19
Notation for Binomial Experiments
Symbol Description
n The number of times a trial is repeated.
p = P (S) The probability of success in a single trial.
q = P (F) The probability of failure in a single trial. (q = 1
x The random variable represents a count of the number of successes in n trials: x = 0, 1, 2, 3, … , n.
Larson & Farber, Elementary Statistics: Picturing the World, 3e 20
Binomial Experiments
Example: Decide whether the experiment is a binomial experiment. If it is, specify the values of n, p, and q, and list the possible values of the random variable x. If it is not a binomial experiment, explain why.
- You randomly select a card from a deck of cards, and note if the card is an Ace. You then put the card back and repeat this process 8 times. This is a binomial experiment. Each of the 8 selections represent an independent trial because the card is replaced before the next one is drawn. There are only two possible outcomes: either the card is an Ace or not. 4 1 52 13 n = 8 p = = 1 1 12 13 13 q = − = x =0,1,2,3,4,5,6,7,
Larson & Farber, Elementary Statistics: Picturing the World, 3e 21
Binomial Experiments
Example: Decide whether the experiment is a binomial experiment. If it is, specify the values of n, p, and q, and list the possible values of the random variable x. If it is not a binomial experiment, explain why.
- You roll a die 10 times and note the number the die lands on.
This is not a binomial experiment. While each trial (roll) is independent, there are more than two possible outcomes: 1, 2, 3, 4, 5, and 6.
Larson & Farber, Elementary Statistics: Picturing the World, 3e 22
Binomial Probability Formula
In a binomial experiment, the probability of exactly x successes in n trials is
Example:
A bag contains 10 chips. 3 of the chips are red, 5 of the chips are white, and 2 of the chips are blue. Three chips are selected, with replacement. Find the probability that you select exactly one red chip.
x n x x n x n x P x C p q n p q n x x
= −^ = −
1 2 P (1) = 3 C 1 (0.3) (0.7)
p = the probability of selecting a red chip 3 0. 10
q = 1 – p = 0. n = 3 x = 1
= 3(0.3)(0.49) = 0.
Larson & Farber, Elementary Statistics: Picturing the World, 3e 23
Binomial Probability Distribution
Example:
A bag contains 10 chips. 3 of the chips are red, 5 of the chips are white, and 2 of the chips are blue. Four chips are selected, with replacement. Create a probability distribution for the number of red chips selected. p = the probability of selecting a red chip 3 0. 10
q = 1 – p = 0. n = 4 x = 0, 1, 2, 3, 4
3 0.
1 0. 2 0.
4 0.
0 0.
x P (x) The binomial probability formula is used to find each probability.
Larson & Farber, Elementary Statistics: Picturing the World, 3e 24
Finding Probabilities
Example: The following probability distribution represents the probability of selecting 0, 1, 2, 3, or 4 red chips when 4 chips are selected.
a.) P (no more than 3) = P (x ≤ 3) = P (0) + P (1) + P (2) + P (3)
3 0.
1 0. 2 0.
4 0.
0 0.
x P (x)
b.) Find the probability of selecting at least 1 red chip.
a.) Find the probability of selecting no more than 3 red chips.
= 0.24 + 0.412 + 0.265 + 0.076 = 0. b.) P (at least 1) = P (x ≥ 1) = 1 – P (0) = 1 – 0.24 = 0. Complement
Larson & Farber, Elementary Statistics: Picturing the World, 3e 28
Geometric Distribution
A geometric distribution is a discrete probability distribution of a random variable x that satisfies the following conditions.
- A trial is repeated until a success occurs.
- The repeated trials are independent of each other.
- The probability of a success p is constant for each trial.
The probability that the first success will occur on trial x is
P (x) = p(q)x^ – 1, where q = 1 – p.
Larson & Farber, Elementary Statistics: Picturing the World, 3e 29
Geometric Distribution
Example:
A fast food chain puts a winning game piece on every fifth package of French fries. Find the probability that you will win a prize,
a.) with your third purchase of French fries,
b.) with your third or fourth purchase of French fries.
p = 0.20 q = 0.
= (0.2)(0.8)^2
a.) x = 3 P (3) = (0.2)(0.8)3 – 1
b.) x = 3, 4 P (3 or 4) = P (3) + P (4)
≈ 0.128 + 0.
Larson & Farber, Elementary Statistics: Picturing the World, 3e 30
Geometric Distribution
Example:
A fast food chain puts a winning game piece on every fifth package of French fries. Find the probability that you will win a prize,
a.) with your third purchase of French fries,
b.) with your third or fourth purchase of French fries.
p = 0.20 q = 0.
= (0.2)(0.8)^2
a.) x = 3
P (3) = (0.2)(0.8)3 – 1
b.) x = 3, 4
P (3 or 4) = P (3) + P (4) ≈ 0.128 + 0.
Larson & Farber, Elementary Statistics: Picturing the World, 3e 31
Poisson Distribution
The Poisson distribution is a discrete probability distribution of a random variable x that satisfies the following conditions.
- The experiment consists of counting the number of times an event, x, occurs in a given interval. The interval can be an interval of time, area, or volume.
- The probability of the event occurring is the same for each interval.
- The number of occurrences in one interval is independent of the number of occurrences in other intervals.
( ) μ x^ eμ P x x!
−
The probability of exactly x occurrences in an interval is
where e ≈ 2.71818 and μ is the mean number of occurrences.
Larson & Farber, Elementary Statistics: Picturing the World, 3e 32
Poisson Distribution
Example:
The mean number of power outages in the city of Brunswick is 4 per year. Find the probability that in a given year,
a.) there are exactly 3 outages,
b.) there are more than 3 outages.
4 (2.71828)^3 -
P =
a .) μ = 4 ,x= 3
b.) P(m or e th a n 3)
= 1 − [ P (3) +P (2) + P (1) + P(0)]
= 1 − P x( ≤3)
