Transcript Chapter 4: Random Variables and Probability Distributions

Statistics
Chapter 4: Discrete Random Variables
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
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Where We’re Going



Develop the notion of a random
variable
Numerical data and discrete random
variables
Discrete random variables and their
probabilities
McClave, Statistics, 11th ed. Chapter 4: Discrete
Random Variables
3
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.
McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
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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
McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
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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
McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
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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
McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
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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
McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
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4.3: Expected Values of
Discrete Random Variables

The mean, or expected value, of a
discrete random variable is
  E( x)   xp( x).
McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
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4.3: Expected Values of
Discrete Random Variables

The variance of a discrete random
variable x is
 2  E[( x  )2 ]  ( x  )2 p( x).

The standard deviation of a discrete
random variable x is
  E[( x   ) ] 
2
2
 (x  )
McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
2
p( x).
10
4.3: Expected Values of
Discrete Random Variables
Chebyshev’s Rule
Empirical Rule
≥0
 .68
P(   2  x    2 )
≥ .75
 .95
P(   3  x    3 )
≥ .89
 1.00
P(     x     )
McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
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4.3: Expected Values of
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%
P( win $1)  .4737
P(lose $1)  .5263
  $1 .4737 $1 .5263 .0526
  .9986
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.)
McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
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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
McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
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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
McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
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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
THH
(q)(p)(p)
p2q
HTT
(p)(q)(q)
pq2
THT
(q)(p)(q)
pq2
TTH
(q)(q)(p)
pq2
TTT
(q)(q)(q)
q3
McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
(3)p2q1
(3)p1q2
(1)p0q3
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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
 n  x n x
P( x)    p q
 x
McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
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4.4: The Binomial Distribution

The Binomial Probability Distribution
 n  x n x
P( x)    p q
 x
McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
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4.4: The Binomial Distribution


Say 40% of the
class is female.
What is the
probability that 6
of the first 10
students walking
in will be female?
 n  x n x
P ( x )    p q
 x
10 6
  (.4 )(.610 6 )
6
 210(.004096)(.1296)
 .1115
McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
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4.4: The Binomial Distribution

A Binomial Random Variable has
Mean
Variance
Standard Deviation
  np
  npq
2
  npq
McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
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4.4: The Binomial Distribution

For 1,000 coin flips,
  np  1000 .5  500
  npq  1000 .5  .5  250
2
  npq  250  16
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%.
McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
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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
McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
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4.5: The Poisson Distribution
P( x) 




x 
e
x!
 = mean number of occurrences in the
given unit of time, area, volume, etc.
e = 2.71828….
µ=
2 = 
McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
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4.5: The Poisson Distribution

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?
P( x  5) 
x 
e
5
3
3e

 .1008
x!
5!
McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
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4.5: The Poisson Distribution

How about in the next 50 yards, assuming a
Poisson distribution?

Since the distance is only half as long,  is only
half as large.
P( x  5) 
x 
e
5
1.5 e

x!
5!
1.5
McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
 .0141
24
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


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.
McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
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4.6: The Hypergeometric
Distribution
 r  N  r 
 

x  n  x 

P( x) 
N
 
n
where
N = the total number of elements
r = number of successes in the N elements
n = number of elements drawn
X = the number of successes in the n elements
McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
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4.6: The Hypergeometric
Distribution
 r  N  r 
 

x  n  x 

P( x) 
N
 
n
nr

N
r ( N  r ) n( N  n)
2
 
N 2 ( N  1)
McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
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4.6: The Hypergeometric
Distribution


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.)
 5 10  5 
 

2  2  2  (10)(1)

P ( M  2)  P ( F  2) 

 .22
45
10
 
2
P ( M  2 or F  2)  P ( M  2)  P ( F  2)  2  .22  .44
McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables
29