Transcript Chapter 7 Random Variables a variable whose value is a numerical

Chapter 7
Random Variables
a variable whose value is a numerical outcome
of a random phenomenon.
7.1 Discrete and Continuous Random Variables
7.2 Means and Variances of Random Variables
26 Discrete &
Continuous,
Mean/Var,
Law of Large
#s
7.2, 7.7, 7.17,
7.24, 7.27,
7.28, 7.32
27
Rules for
Means and
Variances
7.38, 7.47,
7.49, 7.60
28
29
30
REVIEW
REVIEW
CHAPTER 7
TEST
7.1 Discrete and Continuous
Random Variables
Discrete Random Variable
• a random variable with a countable number of outcomes
• the probability distribution lists the values and their
probabilities.
Example: Mrs. Pfeiffer’s Grade Distribution
X = 15% A’s, 30% B’s, 30% C’s, 15% D’s, and 10% F’s
x = Grade
Probability
0
.1
1
.15
2
.30
3
.30
4
.15
Find the probability that a randomly selected student
receives a B or better. P(x ≥ 3) =
7.1 Discrete and Continuous
Random Variables
Continuous Random Variable
• a random variable that takes all values in an interval of
numbers
• the probability distribution is described by a density curve
• the probability of an event is the area under the density curve
Example: Let S = {all numbers 𝑥 such that 0 ≤ 𝑥 ≤ 1}
The result of many trials are represented by the density curve of
a uniform distribution.
0
a. 𝑃(.3 ≤ 𝑥 ≤ .7)
b. 𝑃(𝑥 = .6)
1
c. 𝑃(𝑥 > .5)
Means of Discrete Random
Variables
Mean of a Discrete Random Variable
(aka Expected Value or Weighted Average)
Mean = E(X) = µx =
𝑥𝑖 𝑝𝑖
Example – Suppose you are playing the Missouri
Pick 3 Lotto. If you match the 3 numbers exactly,
you win $500. How much can you expect to win?
Expected Value Explanation
We predict that in the LONG RUN, the average
Lottery ticket buyer wins an average of $0.50
each time. Look at it from the state’s
standpoint: Lotto tickets cost $1, so in the LONG
RUN, the state keeps half of the money
everyone wages.
Expected Value can be misleading – we don’t
expect to win $0.50 on one lottery ticket.
Variances of Discrete Random
Variables
Variance of a Discrete Random Variable
Variance = Var(X) = σx = 𝑥𝑖 − 𝜇𝑥 2 𝑝𝑖
Standard Deviation = σx = sq. root of variance
2
Example – Suppose you are playing the Missouri
Pick 3 Lotto. If you match the 3 numbers exactly,
you win $500. What is the variation of the payout?
Example – Mean and Variance
of a Discrete Random Variable
Toss 4 coins and record the number of heads.
Create a probability distribution table and find
the mean and standard deviation of X.
X=
Example – Mean of a
Continuous Random Variable
Find the value of X for which the area under the
curve is ½ on each side.
𝜇𝑥 = 1
density curve
𝜇𝑥 = 𝜇
normal distribution
Law of Large Numbers
Law of Large Numbers says that for ALL
POPULATIONS (not just normal distributions)
that as the number of observations drawn
increases, the mean of the observed values (the
sample mean = 𝑥) eventually approaches the
mean of the population (𝜇).
This is why casinos stay in business!!!
WARM UP
The distribution of weights of 9 ounce bags of a
particular brand of potato chips is approximately
Normal with mean μ = 9.12 ounces and standard
deviation σ = 0.15 ounce.
a) Draw an accurate sketch of the distribution of potato
chip bag weights. (Be sure to label 1, 2, and 3, standard
deviations from the mean.)
b) A bag that weighs 8.87 ounces is at what percentile in
this distribution?
c) What percent of bags weigh between 8.25 ounces and
9.25 ounces?
d) The top 10% of all bags weigh at least what?
Rules for Means and Variances