AP Statistics
Chapter 7 – Random Variables
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 RV that counts the number of “Tails” in
three tosses of a balanced coin.
• X
• p(x)
______________________________________
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.
Mean Formula
• For a discrete random variable with the
distribution.
X:
P(X):
x1
p1
• μx = ∑ (xi * pi )
x2
p2
x3
p3
x4 . . . . xk
p4. . . . pk
Variance/ Standard Deviation
• The variance of a random variable is
represented by σ2x
• The standard deviation of a random variable is
represented by σx.
• For a discrete random variable:
X:
P(X):
x1
p1
x2
p2
• σ2x = ∑(xi – μx)2 pi
x3
p3
x4 . . . . xk
p4. . . . pk
Law of Large Numbers
• As the sample size increases, the sample mean
approaches the ________________________.
• Holds true for _________________________.
– Draw independent observations at random from any
population with finite mean μ.
– Decide how accurately you would like to estimate the
mean μ.
– As the number of observations drawn increases, the mean
𝑥 of the observed values eventually approaches the mean
μ of the population as closely as you specified and then
stays that close.
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.