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AP Statistics - Chapter 7: Random Variables
Chapter Objectives:
A) explain the differences between discrete and continuous random variables and find simple probabilities for
each
B) calculate means and variances of discrete random variables (and "payout" type problems)
C) calculate means and variances of combined discrete random variables
D) calculate means and variances of combined continuous random variables; use them to calculate
probabilities
Introductory Activity (Crappes)
With a partner, roll two dice and sum:
1) 7, 11 = automatic win
2) 2, 3, 12 = automatic loss
3) any other roll = "point"
a) roll "point" again = win
b) roll 7 = loss
Track all types of wins & losses for 20 games. Track the results for each "point".
A) Discrete & Continuous Random Variables
discrete - has a countable number of values for x; you can list the probabilities for each possible x
EXAMPLE 1:
x
p(x)
P(7 or 11) = _________
P(2, 3, or 12) = ________
continuous - values are spread over an entire interval of numbers
EXAMPLE 2: Calculate probabilites for the following graphs:
A.
1) P(.3 < x < .7) =
2) P(.3 ≤ x ≤ .7) =
3) P(x > .7) =
4) P(x > .3 or x < .7) =
B.
1) P(10 ≤ x ≤ 30) =
C.
1) P(10 < x < 70) =
2) P(x < 10) =
3) P(x > 70) =
4) P(x < 10 or x > 70) =
EXAMPLE 3: Calculate probabilites for the following graph:
a) x =
e) P(x > 1.5) =
b) P(x < 2) =
f) P(2 < x ≤ 7) =
c) P(x = 3) =
g) P(1 < x < 2 or 3 < x < 3.5)=
d) P(1 ≤ x < 3) =
EXAMPLE 4: Heights vary Normally with mean = 50 and s.d. = 10. Find the following probabilities:
a) P(x < 40) =
b) P(x > 55) =
c) P(45 < x < 60) =
Homework: p 470 2, 3, 5, 7-9, 14-18bc
B) Means and Variances of Discrete Random Variables
Probabilities don’t have to be equally likely, so you can’t just add and divide for mean.
Formula to calculate mean: μ = Σ[xi . p(xi)]
Formula to calculate variance: σ2 = Σ [(xi – μ)2 . p(xi)]
standard deviation: σ =
2
Law of Large Numbers - as you obtain more and more observations, the sample mean ( ) eventually approaches the population mean
(.)
The average results of many independent observations are stable and predictable. This helps insurance companies and casinos, etc.
stay in business.
Law of Small Numbers:
list a random sequence of 10 tosses of a coin
psychological – we think we are “due”
NOT true! probability has no memory!
EXAMPLE #5a: Find the average age and standard deviations of age for the class.
list of ages:
mean =
standard deviation =
EXAMPLE 5b: List the ages as groups within a table. Recalculate mean and s.d. using grouped data.
Age
Tally
Prob.
x . p(x)
(x – mean)2 . p(x)
EXAMPLE 6: Find mean and standard deviation for the following:
a.
x
0
10
P(x)
.8
.2
b.
x
1
2
3
4
P(x)
0.1
0.2
0.3
0.4
c.
x
100
500
1000
P(x)
.3
.4
.3
d.
x
1
5
8
14
P(x)
0.15
0.22
0.35
0.28
Homework: p486 25-30, 32-34
AP Statistics - Chapter 7: Random Variables (Page 3)
C) Means and Variances of Combined Discrete Random Variables
Rules for Combining Means and Variances
a+bx
x+y
x-y
ax+by
=
=
=
=
2a+bx
2x+y
2x-y
2ax+by
a + bx
x + y
x - y
ax + by
EXAMPLE 7:
x = 5
x = 4
=
=
=
=
b22x (assumes that x & y are independent)
2x + 2y (assumes that x & y are independent)
2x + 2y (assumes that x & y are independent)
a22x + b22y (assumes that x & y are independent)
y = 10
y = 2
Find: a. x+y
b. x+y
c. x-y
If z = 10x+5y, find:
e. z
f. z
d. x-y
EXAMPLE 8: a) Find mean and standard deviation for each type of vehicle.
Cars are sold as follows:
# of cars
(x)
P(x)
0
0.3
1
0.4
2
0.2
3
0.1
SUVs are sold as follows:
# of SUVs
(y)
P(y)
0
0.4
1
0.5
2
0.1
mean = _____
s.d. = _____
mean = _____
s.d. = _____
b) If you expect to make $350 from each car and $400 from each SUV, what profit will you expect to make? What is the standard
deviation of that profit? For profit, use Z = 350X + 400Y
z = 350x+400y
σ2z = σ2350x + 400y
EXAMPLE 9: Let X and Y be two independent random variables with x = 6, x = 3, y = 7, and y = 2.
a. Find x + y
b. Find x-y
c. Find 3x-4y
C) Means and Variances of Combined Continuous (Normal) Random Variables
Rules for Combining Independent Normal Random Variables - any linear combination of independent Normal random variables is
also Normal.
Normal probability: Normalcdf (lower, upper, , )
AP Statistics - Chapter 7: Random Variables (Page 4)
EXAMPLE 10: Tom and George play golf. Tom’s score (X) varies normally with mean 110 and standard deviation 10. George’s
score (Y) varies normally with mean 100 and standard deviation 8. If their scores are independent:
a) What is the probability that Tom scores below 100?
b) What is the probability that George scores above 110?
c) What is the probability that Tom beats (scores lower than) George?
d) What is the probability that George beats Tom?
e) What is the probability that George beats Tom by more than 10?
f) What is the probability the Tom beats George by more than 15?
Homework: p499 37-52, 55-61
7.2 Homework (page 1)
5. The following gives information about the number of letters in last names. Find the mean and standard deviation for number of
letters in last name.
#Letters
#Students
4
10
5
15
6
15
7
10
6. The chart lists the age (with probabilities) at which a child learns to ride a bike. Find the mean and s.d. for age at which a child
learns to ride.
Age
P(x)
a. What is the prob a child learns to ride by age 6?
3
.05
4
.15
5
.20
b. What is the prob a child learns after age 5?
6
.40
7
.15
8
.05
7. You are playing a lottery with a 10% chance of winning $1000. Find mean and standard deviation of your expected winnings.
How much would you be willing to pay for a ticket?
8. You play a lottery with two prizes: 1st prize = $100, 2nd prize = $50. If you’re 1% likely to win first prize and 5% likely to win
second prize, what are the mean and standard deviation of your expected winnings? Would you be willing to spend $3 for a ticket?
9: You play a lottery. You pick a 3 digit number (all different digits). If your number is picked in exact digit order, you win $500. If
the digits are picked in a different order, you win $100. Find mean and standard deviation of expected winnings.
10. You run a lemonade stand. You expect your sales for the month to be:
#Cups
10
25
40
60
Prob.
0.2
0.3
0.1
0.4
If you make $2 in profit for each cup you sell, what is your expected profit?