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Transcript
Sec. 4.1 Probability Distributions
A ___________ ___________ π‘₯ represents a numerical value associated with
each outcome of a probability experiment.
A random variable is __________ if it has a finite or countable number of possible
outcomes that can be listed.
A random variable is __________ if it has an uncountable number of possible
outcomes, represented by an interval on the number line.
Decide whether or not the random variable is discrete or continuous.
The number of sales calls
The number of hours spent on sales calls
The number of stocks in the Dow Jones Industrial Average
The volume of water in a bucket
The height of a skyscraper
The number of windows in a skyscraper
Discrete Probability Distributions
A ___________ _______________ ______________ lists each possible value the
random variable can assume, together with its probability. A probability
distribution must satisfy the following conditions.
1. The probability of each value of the discrete random variable is between 0 and
1, inclusive, or 0 ≀ 𝑃(π‘₯) ≀ 1.
2. The sum of all the probabilities is 1 or βˆ‘ 𝑃(π‘₯) = 1
Construct a probability distribution in the tables for the following scenarios.
An industrial psychologist administered a personality inventory test for passiveaggressive traits to 150 employees. Individuals were given a score from 1 to 5,
where 1 was extremely passive and 5 extremely aggressive. A score of 3 indicated
neither trait. The results are shown below.
Frequency Distribution
Score, x
1
2
3
4
5
Frequency, 𝑓
24
33
42
30
21
Probability Distribution
A company tracks the number of sales new employees make each day during a
100-day probationary period. The results for one new employee are shown at the
left.
Sales per day, π‘₯
0
1
2
3
4
5
6
7
Number of days, 𝑓
16
19
15
21
9
10
8
2
Probability Distribution
Verify that the following are probability distributions.
Days of rain, π‘₯
0
1
2
3
Probability, 𝑃(π‘₯)
0.216
0.432
0.288
0.064
π‘₯
2
3
4
𝑃(π‘₯)
βˆ’0.1
0.9
0.2
π‘₯
5
6
7
8
𝑃(π‘₯)
1
16
5
8
1
4
1
32
Mean, Variance, and Standard Deviation
The ___________ of a discrete random variable is given by πœ‡ = βˆ‘ π‘₯𝑃(π‘₯).
Note: The mean of the random variable represents the β€œtheoretical average” of a
probability experiment and sometimes is not a possible outcome.
The ____________ of a discrete random variable is 𝜎 2 = βˆ‘(π‘₯ βˆ’ πœ‡)2 𝑃(π‘₯).
The ____________ _____________ is 𝜎 = √𝜎 2 = βˆšβˆ‘(π‘₯ βˆ’ πœ‡)2 𝑃(π‘₯).
Use the tables to find the mean and variance of the probability distributions of
the personality inventory from the previous example.
π‘₯
1
2
3
4
5
π‘₯
1
2
3
4
5
𝑃(π‘₯)
0.16
0.22
0.28
0.20
0.14
𝑃(π‘₯)
0.16
0.22
0.28
0.20
0.14
Find the mean and variance of the probability distribution involving the number
of sales of an employee.
Expected Value
The _______________ ___________ of a discrete random variable is equal to the
mean of the random variable.
Expected Value = 𝐸(π‘₯) = πœ‡ = βˆ‘ π‘₯𝑃(π‘₯)
Note: Expected value is another word for mean in a probability distribution, but
carries a slightly different interpretation. The expected value is what we would
expect to happen over thousands of trials of the same game or probability
experiment.
Find the expected value of the following problems.
At a raffle, 1500 tickets are sold at $2 each for four prizes of $500, $250, $150,
and $75. You buy one ticket. What is the expected value of your gain?
You are playing a game with a friend that involves rolling a normal six-sided die.
You pay $3.50 to play the game. After you roll the die, you earn the same number
of dollars of the number rolled.