Download 7.2 Day 2: Rules for Means and Variances

7.2 Day 2: Rules for
Means and Variances
3/4
CW
Probability WARM UP2/3
1/2
1/3
R
R
NE
A travel agent books passages on three different tours,
1/2Caribbean
1/6
R
with half of her customers
choosing
Waters
EV
(CW), one-third choosing New England’s Historic Trail
(NE), the rest choosing European Vacation (EV). The
agent has noted that three-quarters of those who take CW
return to book passage again, two-thirds of those who
take NE return and one-half of those who take EV return.
If a customer does return, what is the probability that the
person went on NE?
2
16
9

3 2 1 49
 
8 9 12
Rules for Means
Rule 1: If X is a random variable and a and b
are fixed numbers, then
μa + bX = a + bμX
Rule 2: If X and Y are random variables, then
μX + Y = μX + μY
Ex 1: Linda sells cars and trucks
Cars Sold:
0
1
2
3
Probability
0.3
0.4
0.2
0.1
Trucks Sold:
Probability

0
1
2
0.4
0.5
0.1
Find the mean of each of these random variables.
μx = 1.1 cars
μy = 0.7 trucks

At her commision rate of 25%, Linda expects
to earn $350 for each car sold and $400 for
each truck sold. So her earnings are
Z = 350X + 400Y

Combining rules 1 and 2, her mean earnings are
μz = 350μx + 400μy
= (350)(1.1) + (400)(0.7) = $650
Rules for Variances
Adding a constant a
to a random variable
changes its mean but
does not change its
variability.
Rule 1: If X is a random variable and a and b
are fixed numbers, then
σ2a + bX = b2σx2
Rule 2: If X and Y are independent random
variables, then
σ2X+Y = σx2 + σY2
σ2X – Y = σx2 + σY2
This is the addition rule for variances of
independent random variables.
Rules for Standard Deviations


Standard deviations are most easily
combined by using the rules for variances
rather than by giving separate rules for
standard deviations.
Note that variances of independent variables
add, standard deviations do not generally
add!
Ex 2: Winning the lottery
Recall that the payoff X of a Tri-State lottery
ticket is $500 with probability 1/1000 and $0
the rest of the time.
a) Calculate the mean and variance.
xi
pi
xipi
(xi – μx)2pi
0
0.999
0
0.24975
500
0.001
0.5
249.50025
μx = 0.5
σx2 = 249.75
Games of chance
typically have large
standard deviations.
Large variability makes
gambling more
exciting!
b) Find the standard deviation.
σx = $15.80
Basically,
you lose an
average of
50 cents on
a ticket.
c) If it cost $1 to buy a ticket, what is the mean
amount that you win?
μw = μx – 1 = -$0.50
Note that the variance and standard deviation stay
the same by Rule 1 for Variances.
d) Suppose that you buy two tickets on two
different days. These tickets are independent
due to the fact that drawings are held each day.



Find the mean total payoff X + Y.
Note that this is not the
μX + Y = μX + μY = $0.50 + $0.50
= $1.00
sum of the individual
standard deviations
Find the variance of X + Y.
($15.80 + $15.80)
σ2X + Y = σX2 + σY2 = 249.75 + 249.75 = 499.50
Find the standard deviation of the total payoff.
σX + Y = 499.5  $22.35
Ex 3: SAT Scores

Below are the means and standard
deviations of SAT scores at a certain college.
SAT Math Score X
SAT Verbal Score Y

It wouldn’t make
sense to add the
standard deviations,
due to the fact that
the test scores are
not independent.
μx = 519 σx = 115
μY = 507 σY = 111
Find the mean overall SAT Score.
μX + Y = μX + μY = 519 + 507 = 1026