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The probability model describes the number of repair
calls that an appliance repair shop may receive during an
hour.
a)How many calls should the shop expect per hour?
b)What is the standard deviation?
Repair Calls 0 1 2 3
Probability 0.1 0.3 0.4 0.2
A man buys a racehorse for $20,000 and enters it in two
races. He plans to sell the horse afterward, hoping to
make a profit. If the horse wins both races, its value will
jump to $100,000. If it wins one of the races, it will be
worth $50,000. If it loses both races, it will be worth only
$10,000. The man believes there’s a 20% chance that the
horse will win the first race and a 30% chance it will win
the second one. Assuming that the two races are
independent events, find the man’s expected profit.
Chapter 16 Part 3
MORE ABOUT MEANS AND VARIANCES
Adding a constant to data (adding the same amount
to each value) shifts the mean but the variance and
standard deviation do not change.
𝐸 𝑋±𝑐 =𝐸 𝑋 ±𝑐
𝑉𝑎𝑟 𝑋 ± 𝑐 = 𝑉𝑎𝑟 𝑋
𝑆𝐷 𝑋 ± 𝑐 = 𝑆𝐷 𝑋
Example:
Couples at the Quiet Nook can expect discounts averaging $5.83
with a standard deviation of $8.62. Suppose that for several
weeks the restaurant has also been distributing coupons worth
$5 off any one meal per table. If every couple there brings a
coupon, what will be the mean and standard deviation of the total
discounts they’ll receive?
E(x) = 5.83 + 5 = $10.83
SD(x) = $8.62 (unchanged)
Multiplying each value of a random variable by
a constant multiplies the mean and the
standard deviation by that constant.
𝐸 𝑎𝑋 = 𝑎 ∙ 𝐸(𝑋)
𝑆𝐷 𝑎𝑋 = 𝑎 ∙ 𝑆𝐷(𝑋)
Example: (continued)
Couples at the Quiet Nook can expect discounts averaging
$5.83 with a standard deviation of $8.62. When two
couples dine together on a single check, the restaurant
doubles the discount offer - $40 for the ace of hearts on
the 1st draw or $20 for ace of hearts on 2nd draw. What are
the mean and standard deviation?
E(x) = 2(5.83) = $11.66
StandardDev(x) = 2(8.62) = $17.24
Example:
Given independent random variables with means and
standard deviations as shown, find the mean and
standard deviation of:
Standard
Mean
2Y + 20
Deviation
Mean = E(2Y+20) =
2E(Y)+20 = 2(12)+20 = 44
X
80
12
SD = SD(2Y+20) = 2SD(Y)
= 2(3) = 6
Y
12
3
Adding/Subtracting Random Variables:
𝐸 𝑋 ± 𝑌 = 𝐸(𝑋) ± 𝐸(𝑌)
If X and Y are independent, then
𝑉𝑎𝑟 𝑋 ± 𝑌 = 𝑉𝑎𝑟 𝑋 + 𝑉𝑎𝑟 𝑌
𝑆𝐷 𝑋 ± 𝑌 =
𝑆𝐷(𝑋)2 + 𝑆𝐷(𝑌)2
Example:
Given independent random variables with means and
standard deviations as shown, find the mean and
standard deviation of:
Standard
Mean
X+Y
Deviation
Mean = E(X+Y) =
E(X)+E(Y)= 80+12= 92
X
80
12
SD = SD(X+Y)
= 122 + 32 = 12.37
Y
12
3
Practice:
Given independent random variables with means and
standard deviations as shown, find the mean and
standard deviation of:
Standard
a) 3X
b) 0.25X+Y
c) X – 5Y
d) 𝑋1 +𝑋2 + 𝑋3
Mean
Deviation
X
80
12
Y
12
3
Answers:
a)240; 36
b)32; 4.24
c)20; 19.21
d)240; 20.78
Today’s Assignment:
Add to HW: p. 384 #24-26, 33-36
Chapter 16 HW Due Monday 12-7-15