# 4.1-4.2: Random Variables

```CHS Statistics
Objective: Use experimental and theoretical
distributions to make judgments about the
likelihood of various outcomes in uncertain
situations
Warm-Up

 Decide if the following random variable x is discrete(D) or
continuous(C).
1) X represents the number of eggs a hen lays in a day.
2) X represents the amount of milk a cow produces in one day.
3) X represents the measure of voltage for a smoke-detector
battery.
4) X represents the number of patrons attending a rock concert.
Random Variable X

 Random variable - A variable, usually denoted as x,
that has a single numerical value, determined by
chance, for each outcome of a procedure.
 Probability distribution – a graph, table, or formula
that gives the probability for each value of the
random variable.
Random Variable X

 A study consists of randomly selecting 14
newborn babies and counting the number of
girls. If we assume that boys and girls are
equally likely and we let x = the number of girls
among 14 babies…
 What is the random variable?
 What are the possible values of the random
variable (x)?
 What is the probability distribution?
Probabilities of
Girls
x (Girls) P(x)
0
0
1
0.001
2
0.006
3
0.022
4
0.061
5
0.122
6
0.183
7
0.209
8
0.183
9
0.122
10
0.061
11
0.022
12
0.006
13
0.001
14
0
Types of Random Variables

 A discrete random variable has either a finite
number of values or a countable number of values.
 A continuous random variable has infinitely many
values, and those values can be associated with
measurements on a continuous scale in such a ways
that there are no gaps or interruptions.
 Usually has units
Discrete Probability Distributions

 A Discrete probability distribution lists each possible
random variable value with its corresponding probability.
 Requirements for a Probability Distribution:
1. All of the probabilities must be between 0 and 1.
 0 ≤ P(x) ≤ 1
2. The sum of the probabilities must equal 1.
 ∑ P(x) = 1
Discrete Probability Distributions (cont.)

 The following table represents a probability distribution.
What is the missing value?
x
1
2
3
4
P(x)
0.16
0.22
0.28
0.2
5
Discrete Probability Distributions (cont.)

 Do the following tables represent discrete probability distributions?
1)
4)
2)
x
P(x)
0.216
5
2
0.432
3
4
x
P(x)
0
3)
x
P(x)
0.28
1
1/2
6
0.21
2
1/4
0.288
7
0.43
3
5/4
0.064
8
0.15
4
-1
x
P(x)
1
.09
2
0.36
3
0.49
4
0.06
5) P(x) = x/5, where x can be 0,1,2,3
6) P(x) = x/3, where x can be 0,1,2
Mean and Standard Deviation
of a Probability Distribution
 Mean: 𝝁 = [𝒙 ∙ 𝑷 𝒙 ]
 Standard Deviation: 𝝈 =

[ 𝒙𝟐 ∙ 𝑷(𝑿)] − 𝝁𝟐
 Calculator:
 Calculate as you would for a weighted mean or frequency
distribution:
Very important!





Stat  Edit
L1 = x values
L2 = P(x) values
Stat  Calc
1: Variable Stats L1, L2
Mean and Standard Deviation of a
Probability Distribution (cont.)

 Calculate the mean and standard deviation of the following
probability distributions:
1) Let x represent the # of games
required to complete the World Series:
2) Let x represent the # dogs per household:
X = # of
Dogs
Households
0
1491
0.253
1
425
6
0.217
2
168
7
0.410
3
48
x
P(x)
4
0.480
5
Expected Value

 The expected value of a discrete random variable
represents the average value of the outcomes, thus is
the same as the mean of the distribution.
𝑬=𝝁=
[𝒙 ∙ 𝑷 𝒙 ]
Expected Value

 Consider the numbers game, often called “Pick Three” started many years
ago by organized crime groups and now run legally by many
governments. To play, you place a bet that the three-digit number of your
choice will be the winning number selected. The typical winning payoff is
499 to 1, meaning for every \$1 bet, you can expect to win \$500. This leaves
you with a net profit of \$499. Suppose that you bet \$1 on the number 327.
What is your expected value of gain or loss? What does this mean?
Event
Win
Lose
x
P(x)
Assignment

 pp. 190 # 2 – 14 Even, 18 – 22 Even
```