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1
Discrete and Continuous Random Variables:
A __________ is a quantity whose value changes.
A __________ __________ is a variable whose value is obtained by __________.
number of students present
number of red marbles in a jar
number of heads when flipping three coins
students’ grade level
Examples:
A __________ __________ is a variable whose value is obtained by __________.
height of students in class
weight of students in class
time it takes to get to school
distance traveled between classes
Examples:
A __________ __________ is a variable whose value is a numerical outcome of a random
phenomenon.
▪
A random variable is denoted with a ____________________
▪
The __________ __________ of a random variable X tells what the possible values
of X are and how probabilities are assigned to those values
▪
A random variable can be __________ or __________
A __________ __________ __________ X has a countable number of possible values.
Example: Let X represent the sum of two dice.
Then the probability distribution of X is as follows:
X
2
3
4
5
6
7
8
9
10
11
12
P(X)
1
36
2
36
3
36
4
36
5
36
6
36
5
36
4
36
3
36
2
36
1
36
To graph the probability distribution of a __________ __________ __________, construct
a ____________________.
2
Probability Distribution of X
Probability
0.20
0.15
0.10
0.05
0.00
2
3
4
5
6
7
8
9
10
11
12
Outcome
A __________ __________ __________ X takes all values in a given interval of numbers.
▪
The probability distribution of a continuous random variable is shown by a
__________ __________ .
▪
The probability that X is between an interval of numbers is the __________ under
the density curve between the interval endpoints
Means and Variances of Random Variables:
The __________ of a discrete random variable, X, is its ____________________. Each
value of X is weighted by its probability.
To find the mean of X, __________ each value of X by its probability, then __________ all
the products.
μ X = x1 p1 + x2 p2 + ⋅⋅⋅ + xk pk
= ∑ xi pi
The mean of a random variable X is called the __________ __________ of X.
Law of Large Numbers:
As the number of observations increases, the mean of the ____________________, x ,
approaches the mean of the __________, μ .
The more __________ in the outcomes, the more trials are needed to ensure that
close to μ .
x is
3
Rules for Means:
If X is a random variable and a and b are fixed numbers, then
If X and Y are random variables, then
Example:
Suppose the equation Y = 20 + 100X converts a PSAT math score, X, into an SAT math
score, Y. Suppose the average PSAT math score is 48. What is the average SAT math
score?
μ X = 48
μ a +bX = a + bμ X
μ 20+100 X = 20 + 100 μ X
= 20 + 100 ( 48 )
= 500
Example:
Let
Let
μ X = 625
μY = 590
represent the average SAT math score.
represent the average SAT verbal score.
μ X +Y = μ X + μY represents the average combined SAT score. Then
μ X +Y = μ X + μY = 625 + 590 = 1215 is the average combined total SAT score.
The Variance of a Discrete Random Variable:
If X is a discrete random variable with mean μ , then the __________ of X is
σ 2 = ( x1 − μ X ) p1 + ( x2 − μ X ) p2 + ⋅⋅⋅ + ( xk − μ X ) pk
2
2
2
X
= ∑ ( xi − μ X ) pi
2
The standard deviation __________ is the __________ of the __________.
4
Rules for Variances:
If X is a random variable and a and b are fixed numbers, then
If X and Y are independent random variables, then
Example:
Suppose the equation Y = 20 + 100X converts a PSAT math score, X, into an SAT math
score, Y. Suppose the standard deviation for the PSAT math score is 1.5 points. What is
the standard deviation for the SAT math score?
σ X2 = (1.5 ) = 2.25
2
σ a2+bX = b 2σ X2
σ 202 +100 X = (100 ) σ X2
2
= (100 ) ( 2.25 )
2
= 22, 500
σ X2 = 150
Suppose the standard deviation for the SAT math score is 150 points, and the standard
deviation for the SAT verbal score is 165 points. What is the standard deviation for the
combined SAT score?
*** Because the SAT math score and SAT verbal score are not __________, the rule for
adding __________ does not apply!