My Best Idea for Teaching a
Tough Concept: Sums and
Differences of Random Variables
Josh Tabor
joshtabor@hotmail.com
What I hope to show…
1.
2.
It is better to start with an interesting
question than a boring formula
Simulation is a powerful tool for estimation—
and discovery
Speed Dating
Suppose that the height M of male speed
daters follows a Normal distribution with a
mean of 69.5 inches and a standard deviation
of 4 inches and the height F of female speed
daters follows a Normal distribution with a
mean of 65 inches and a standard deviation of
3 inches. What is the probability that a
randomly selected male speed dater is taller
than the randomly selected female speed dater
that he is paired with?
Example from the Annotated Teacher’s Edition for The Practice of
Statistics 4e by Starnes, Yates, and Moore
Speed Dating

On your calculators, generate the height of a
randomly selected Male speed dater


On your calculators, generate the height of a
randomly selected Female speed dater


TI-84: Math: Prb: RandNorm(69.5,4)
TI-84: Math: Prb: RandNorm(65,3)
Who is taller??
Speed Dating




Now, we will each generate 100 speed dating
couples (well, we can at least generate their
heights…)
In the heading of L1, enter
RandNorm(69.5,4,100)
In the heading of L2, enter
RandNorm(65,3,100)
What should we do next?
Speed Dating



How can we estimate P(M – F > 0) without
doing a simulation?
We need to know the distribution of M – F.
To Fathom!
Speed Dating
Screen Shot (just in case…)
Speed Dating
Screen Shot (just in case…)
Speed Dating




What did we learn about the distribution of
M – F?
Shape: Approximately Normal
Center: Mean = 69.5 – 65 = 4.5
Spread: SD = 42 + 32 = 5
Speed Dating



Rules for the difference of two independent
random variables, X and Y:
Mean: 𝜇𝑋−𝑌 = 𝜇𝑋 − 𝜇𝑌
SD: σ=
X −Y

σ +σ
2
X
2
Y
Note: Just like the Pythagorean Theorem only works for right
triangles, this formula only works when the variables are
independent.
Speed Dating…at a Circus?




What do we know about the sum of two
independent random variables?
That is, what if a randomly selected female
were to stand on the head of a randomly
selected male?
What is the distribution of M + F?
To Fathom!
Speed Dating…at a Circus?

Screen Shot (just in case…)
Speed Dating…at a Circus?



Rules for the sum of two independent random
variables, X and Y:
Mean: 𝜇𝑋+𝑌 = 𝜇𝑋 + 𝜇𝑌
SD: σ=
X +Y

σ +σ
2
X
2
Y
Note: Just like the Pythagorean Theorem only works for right
triangles, this formula only works when the variables are
independent.
Extension 1



From a student: What about the product of
two random variables?
Not a standard topic in intro stats!
To Fathom!
Extension 1
Screen Shot (just in case…)
Extension 1

Rules for the product of two independent
random variables, X and Y:

Mean:

SD:

µ XY = µ X µY
σ XY =
σ σ +σ µ +σ µ
2
X
2
Y
2
X
2
Y
2
Y
2
X
SD formula courtesy of Dr. Roy St. Laurent, Northern Arizona University
Extension 2


What about the sum of more than 2 random
variables?
Suppose that the weights of apples are
Normally distributed with a mean of 6 ounce
and a standard deviation of 1 ounce. If you
were to repeatedly select 9 apples at random
and find the sum of their weights, what are
the mean and standard deviation of the sum?
Extension 2
Shape: Should be approximately Normal
Mean = 6+6+6+6+6+6+6+6+6 = 54
SD = 12 + 12 + 12 + 12 + 12 + 12 + 12 + 12 + 12 =
9= 3
Extension 2
Screen Shot (just in case…)
Thanks!
Have questions?
Want the powerpoint?
Email me: joshtabor@hotmail.com