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AP Statistics
Chapter 7 Notes
Random Variables
Random Variable
– A variable whose value is a numerical
outcome of a random phenomenon.
Discrete Random Variable
– Has a countable number of outcomes
– e.g. Number of boys in a family with 3 children
(0, 1, 2, or 3)
Probability Distribution
Lists the values of a discrete random
variable and their probabilities.
Value of X:
P(X) :
x1
p1
x2
p2
x3
p3
x4 . . . . xk
p4. . . . pk
Example of a Probability
Distribution (Discrete RV)
Xage when male college students
began to shave regularly.
X 11 12 13 14 15 16 17 18 19
p(x).013 0
20
.027 .067 .213 .267 .240 .093 .067 .013
Continuous Random Variable
Takes on all values in an interval of
numbers.
– e.g. women’s heights
– e.g. arm length
Probability Distribution for Continuous RV
– Described by a density curve.
– The probability of an event is the area under a
density curve for a given interval.
– e.g. a Normal Distribution
Mean
The mean of a random variable is
represented by μx, μy, etc.
The mean of X is often called the expected
value of X.
– The “expected value” does not have to be a
number that can possibly be obtained,
therefore you can’t necessarily “expect” it to
occur.
Mean Formula
For a discrete random variable with the
distribution.
X: x1
P(X): p1
μx = ∑ xi pi
x2
p2
x3
p3
x4 . . . . xk
p4. . . . pk
Example of a Probability
Distribution (Discrete RV)
Xage when male college students
began to shave regularly.
X 11 12 13 14 15 16 17 18 19
p(x).013 0
20
.027 .067 .213 .267 .240 .093 .067 .013
Variance/ Standard Deviation
The variance of a random variable is
represented by σ2x and the standard
deviation by σx.
For a discrete random variable…
σ2x = ∑(xi – μx)2 pi
Law of Large Numbers
As the sample size increases, the sample
mean approaches the population mean.
Rules for means of Random
Variables
1.μa+bx = a + bμx
– If you perform a linear transformation on
every data point, the mean will change
according to the same formula.
2. μX ± Y = μX ± μY
– If you combine two variables into one
distribution by adding or subtracting, the
mean of the new distribution can be
calculated using the same operation.
Rules for variances of Random
Variables
1. σ2a + bx = b2σ2x
2. σ2X + Y = σ2X + σ2Y
σ2X - Y = σ2X + σ2Y
– X and Y must be independent
Any linear combination of independent
Normal random variables is also Normally
distributed.