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Transcript of Chapter 4: Discrete Random Variables Statistics. McClave, Statistics, 11th ed. Chapter 4: Discrete...
Chapter 4: Discrete Random Variables
Statistics
McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
2
Where We’ve Been
Using probability to make inferences about populations
Measuring the reliability of the inferences
McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
3
Where We’re Going
Develop the notion of a random variable
Numerical data and discrete random variables
Discrete random variables and their probabilities
4.1: Two Types of Random Variables
A random variable is a variable hat assumes numerical values associated with the random outcome of an experiment, where one (and only one) numerical value is assigned to each sample point.
4McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
4.1: Two Types of Random Variables
A discrete random variable can assume a countable number of values. Number of steps to the top of the Eiffel Tower*
A continuous random variable can assume any value along a given interval of a number line. The time a tourist stays at the top
once s/he gets there
*Believe it or not, the answer ranges from 1,652 to 1,789. See Great Buildings
5McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
4.1: Two Types of Random Variables
Discrete random variables Number of sales Number of calls Shares of stock People in line Mistakes per page
Continuous random variables Length Depth Volume Time Weight
6McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
4.2: Probability Distributions for Discrete Random Variables
The probability distribution of a discrete random variable is a graph, table or formula that specifies the probability associated with each possible outcome the random variable can assume. p(x) ≥ 0 for all values of x p(x) = 1
7McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
4.2: Probability Distributions for Discrete Random Variables
Say a random variable x follows this pattern: p(x) = (.3)(.7)x-1
for x > 0. This table gives the
probabilities (rounded to two digits) for x between 1 and 10.
x P(x)
1 .30
2 .21
3 .15
4 .11
5 .07
6 .05
7 .04
8 .02
9 .02
10 .01
8McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
4.3: Expected Values of Discrete Random Variables
The mean, or expected value, of a discrete random variable is
( ) ( ).E x xp x
9McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
4.3: Expected Values of Discrete Random Variables
The variance of a discrete random variable x is
The standard deviation of a discrete random variable x is
2 2 2[( ) ] ( ) ( ).E x x p x
2 2 2[( ) ] ( ) ( ).E x x p x
10McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
)33(
)22(
)(
xP
xP
xP
Chebyshev’s Rule Empirical Rule
≥ 0 .68
≥ .75 .95
≥ .89 1.00
11McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
4.3: Expected Values of Discrete Random Variables
4.3: Expected Values of Discrete Random Variables
12McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
In a roulette wheel in a U.S. casino, a $1 bet on “even” wins $1 if the ball falls on an even number (same for “odd,” or “red,” or “black”).
The odds of winning this bet are 47.37%
9986.
0526.5263.1$4737.1$
5263.)1$(
4737.)1$(
loseP
winP
On average, bettors lose about a nickel for each dollar they put down on a bet like this.(These are the best bets for patrons.)
4.4: The Binomial Distribution
A Binomial Random Variable n identical trials Two outcomes: Success or Failure P(S) = p; P(F) = q = 1 – p Trials are independent x is the number of Successes in n trials
13McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
4.4: The Binomial Distribution
A Binomial Random Variable n identical trials Two outcomes: Success
or Failure P(S) = p; P(F) = q = 1 – p Trials are independent x is the number of S’s in n
trials
Flip a coin 3 times
Outcomes are Heads or Tails
P(H) = .5; P(F) = 1-.5 = .5
A head on flip i doesn’t change P(H) of flip i + 1
14McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
4.4: The Binomial Distribution
Results of 3 flips Probability Combined Summary
HHH (p)(p)(p) p3 (1)p3q0
HHT (p)(p)(q) p2q
HTH (p)(q)(p) p2q (3)p2q1
THH (q)(p)(p) p2q
HTT (p)(q)(q) pq2
THT (q)(p)(q) pq2 (3)p1q2
TTH (q)(q)(p) pq2
TTT (q)(q)(q) q3 (1)p0q3
15McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
4.4: The Binomial Distribution
The Binomial Probability Distribution p = P(S) on a single trial q = 1 – p n = number of trials x = number of successes
xnxqpx
nxP
)(
16McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
4.4: The Binomial Distribution
The Binomial Probability Distribution
xnxqpx
nxP
)(
17McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
Say 40% of the class is female.
What is the probability that 6 of the first 10 students walking in will be female?
4.4: The Binomial Distribution
1115.
)1296)(.004096(.210
)6)(.4(.6
10
)(
6106
xnxqpx
nxP
18McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
4.4: The Binomial Distribution
Mean
Variance
Standard Deviation
A Binomial Random Variable has
McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
19
2
np
npq
npq
4.4: The Binomial Distribution
16250
2505.5.1000
5005.10002
npq
npq
np
For 1,000 coin flips,
The actual probability of getting exactly 500 heads out of 1000 flips is just over 2.5%, but the probability of getting between 484 and 516 heads
(that is, within one standard deviation of the mean) is about 68%.
20McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
4.5: The Poisson Distribution
Evaluates the probability of a (usually small) number of occurrences out of many opportunities in a … Period of time Area Volume Weight Distance Other units of measurement
21McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
4.5: The Poisson Distribution
!)(
x
exP
x
= mean number of occurrences in the given unit of time, area, volume, etc.
e = 2.71828…. µ = 2 =
22McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
4.5: The Poisson Distribution
1008.!5
3
!)5(
35
e
x
exP
x
Say in a given stream there are an average of 3 striped trout per 100 yards. What is the probability of seeing 5 striped trout in the next 100 yards, assuming a Poisson distribution?
23McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
4.5: The Poisson Distribution
0141.!5
5.1
!)5(
5.15
e
x
exP
x
How about in the next 50 yards, assuming a Poisson distribution? Since the distance is only half as long, is only
half as large.
24McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
4.6: The Hypergeometric Distribution
In the binomial situation, each trial was independent. Drawing cards from a deck and replacing
the drawn card each time If the card is not replaced, each trial
depends on the previous trial(s). The hypergeometric distribution can be
used in this case.
McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
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4.6: The Hypergeometric Distribution
McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
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Randomly draw n elements from a set of N elements, without replacement. Assume there are r successes and N-r failures in the N elements.
The hypergeometric random variable is the number of successes, x, drawn from the r available in the n selections.
4.6: The Hypergeometric Distribution
McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
27
n
N
xn
rN
x
r
xP )(
where N = the total number of elementsr = number of successes in the N elementsn = number of elements drawnX = the number of successes in the n elements
4.6: The Hypergeometric Distribution
McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
28
n
N
xn
rN
x
r
xP )(
)1(
)()(2
2
NN
nNnrNrN
nr
4.6: The Hypergeometric Distribution
44.22.2)2()2()2or2(
22.45
)1)(10(
2
10
22
510
2
5
)2()2(
FPMPFMP
FPMP
Suppose a customer at a pet store wants to buy two hamsters for his daughter, but he wants two males or two females (i.e., he wants only two hamsters in a few months)
If there are ten hamsters, five male and five female, what is the probability of drawing two of the same sex? (With hamsters, it’s virtually a random selection.)
McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
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