Download Statistical Data Analysis: Primer

Today
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Today: Finish Chapter 4, Start Chapter 5
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Reading:
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Chapter 5 (not 5.12)
Important Sections From Chapter 4
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4-1-4.4 (excluding the negative hypergeometric distribution)
4.6
Suggested problems: 5.1, 5.2, 5.3, 5.15, 5.25, 5.33, 5.38, 5.47, 5.53, 5.62
Hypergeometric Distribution
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When M/N is essentially constant, the hypergeometric probabilities
can be approximated by using the binomial distribution
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Example
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Suppose 40% of voters of the 500,000 voters in a city are Democrats
A poll of 500 voters is done
What is the probability that 50% of voters claim to be Democrats
Example
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In the game Monopoly, where players roll two dice, a player can end
up in “jail”
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To get out of jail, the player must roll two of a kind to get out of jail
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Find the probability that a player rolls a “doubles” on their turn
Example
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If Z is the random variable denoting the number of turns required to
get out of jail, what is the probability function for Z
Geometric Distribution
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If Z is the number of independent Bernoulli trials (Ber(p)) required to
get a success, then Z has a geometric distribution (Z~Geo(p)),
Geometric Distribution
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Mean:
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Variance:
Example
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In Monopoly, what is the expected number of turns required to get
out of jail?
Example
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Suppose an archer hits a bull’s-eye once in every 10 tries on average
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Find the probability she hits her first bull’s-eye on the 11 trial
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Find the probability she hits her third bull’s-eye on the 15 trial
Negative Binomial Distribution
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If W is the number of independent Bernoulli trials (Ber(p)) required
to get the rth success, then W has a negative binomial distribution,
Geometric Distribution
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Mean:
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Variance:
Example
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Suppose an archer hits a bull’s-eye once in every 10 tries on average
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Find the probability she hits her third bull’s-eye on the 15 trial
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Find the expected number of trials required to get the third bull’s-eye
Example
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Suppose that typographical errors occur at a rate of ½ per page
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Find the probability of getting 3 mistakes in a given page
Poisson Distribution
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If X is a random variable denoting the number (the count) of events
in any region of fixed size, and λ is the rate at which these events
occur, then the probability function for X is:
Example
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Suppose that typographical errors occur at a rate of ½ per page
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Find the probability of getting 3 mistakes in a given page
Example
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Find the expected number of errors on a given page
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What is the probability distribution of the number of errors in a 20
page paper?
Example
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A study on the number of calls to a wrong number at a payphone in a
large train terminal was conducted (Thornedike, 1926)
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According to the study, the number of calls to wrong numbers in a
one minute interval follows a Poisson distribution with parameter
λ=1.20
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Find the probability that the number of wrong numbers in a 1 minute
interval is two
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Find the probability that the number of wrong numbers in a 1 minute
interval is between two and 4
Chapter 5
Continuous Random Variables
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Not all outcomes can be listed (e.g., {w1, w2, …,}) as in the case of
discrete random variable
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Some random variables are continuous and take on infinitely many
values in an interval
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E.g., height of an individual
Continuous Random Variables
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Axioms of probability must still hold
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Events are usually expressed in intervals for a continuous random
variable
0  P( E )  1; for any event E
P()  1
P( E )  P( F )  P( E )  P( F ) whenever E and F are mutually exclusive
Example
(Continuous Uniform Distribution)
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Suppose X can take on any value between –1 and 1
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Further suppose all intervals in [-1,1] of length a have the same
probability of occurring, then X has a uniform distribution on (-1,1)
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Picture:
Distribution Function of a Continuous Random Variable
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The distribution function of a continuous random variable X is
defined as,
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Also called the cumulative distribution function or cdf
Properties
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Probability of an interval:
Example
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Suppose X~U(-1,1), with cdf F(x)=1/2(x+1) for –1<x<1
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Find P(X<0)
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Find P(-.5<X<.5)
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Find P(X=0)
Example
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Suppose X has cdf,
 x / 3, if 0  x  1
F ( x)  
( x  1) / 3, if 1  x  2
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Find P(X<1/2)
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Find P(.5<X<3)
Distribution Functions and Densities
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Suppose that F(x) is the distribution function of a continuous random
variable
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If F(x) is differentiable, then its derivative is:
f ( x)  F ' ( x) 
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d
F ( x)
dx
f(x) is called the density function of X
Distribution Functions and Densities
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Therefore,
a
F (a) 
 f ( x)dx

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That is, the probability of an interval is the area under the density
curve
Example
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Suppose X~U(0,1), with cdf F(x)=x for –1<x<1
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What is the desnity of X?
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Find P(X<.33)
Properties of the Density