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Discrete Distributions

Tieming Ji

Fall 2012

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Definition: A random variable is discrete if it can

take a countable number of values.

Example 1: Flip a coin. If head, X= 1; otherwise, X = 0.

Example 2: Roll a dice. Variable X= outcome.

X = {1, 2, , 6}.Example 3: Flip a coin, and stop when you get a tail. LetXbe the number of flips. Then,the sample space S=

{T, HT, HHT, HHHT,

}, and

the random variable X = {1, 2, 3, 4, }.

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Definition: The probability function (or theprobability mass function, or the probability density

function) for a discrete variable X, f(x), is afunction to describe the probability that X =x, i.e.

f(x) =P(X =x),

such that, for any realization X =x,

1.f(x) 0; and2.all x

f(x) = 1.

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Example 1. Flip a coin with probability pto get a head. Let X = 1,if a head; otherwise X= 0. We have X = {0, 1}, and

f(1) =P(X= 1) = p,

f(0) =P(X = 0) = 1 p,f(1) +f(0) = 1.

Example 2. Flip a coin with probability pto get a head. Stop whenyou get a tail. Let Xdenote the number of total flips. We haveX = {1, 2, 3, }, and

f(k) =P(X =k) =pk1(1

p), k= 1, 2,

k=1

f(k) =

k=1

P(X=k) =

k=1

pk1(1 p) = 1.

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Theorem: (Convergence of geometric series)

k=1

ark1 = a

1 r, where|r|

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Theorem: (Sum of first K terms)

Kk=1

ark1 =a(1 rK)

1 r , where r= 1.

Exercise. Compute

k=113 (1 13 )k. Correctly define the first term

a, and the ratio r.

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Definition: The cumulative distribution function for

random variable X is denoted by F, and defined as

F(x) =P(X x), for real x.

Example 1. Roll a fair dice. Let X= outcome. Compute F(4).

Example 2. Flip a coin with probability p landing a head. Stop whenyou get a tail. Let Xdenote the number of total flips. ComputeF(10).

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Definition: The expectation (or mean) of a discreterandom variable X, E(X), is defined as

E(X) =all x

xf(x).

Extension: The expectation of a function of adiscrete random variable h(X) is denoted byE(h(X)), and computed as

E(h(X)) =all x

h(x)f(x).

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Example 3.3.1 (on page 53). Let Xdenote the number of heartbeats per

minute obtained per patient in a hospital.

x 40 60 68 70 72 80 100f(x) 0.01 0.04 0.05 0.80 0.05 0.04 0.01

What is E(X)?

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Properties:X and Yare random variables, and c is

a constant. Then,(1) E(c) =c,(2) E(cX) =cE(X),(3) E(X+Y) =E(X) +E(Y).

Example 3.3.2 (on page 54). We know E(X) = 7 and E(Y) = 5,compute E(4X 2Y+ 6).

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Definition: The variance of a random variable X,Var(X), is defined as

Var(X) =2X =E(X E(X))2.

Extension 1:

Var(X) =E(X E(X))2 =E(X2) (E(X))2.

Extension 2: The variance of a function of a random

variable h(X) is

Var(h(X)) =2h(X)=E(h(X) E(h(X)))2.

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Example 3.3.3 (page 54). Let X and Ydenote the number of heartbeatsper minute for two groups of patients, respectively. Compute the

expectations and variances for these two variables.

x 40 60 68 70 72 80 100f(x) 0.01 0.04 0.05 0.80 0.05 0.04 0.01

y 40 60 68 70 72 80 100f(y) 0.40 0.05 0.04 0.02 0.04 0.05 0.40

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Properties:X and Yare random variables, c is a

constant. Then(1) Var(c) = 0,(2) Var(cX) =c2Var(X),

(3) IfX and Yare independent, thenVar(X+Y) = Var(X) + Var(Y).

Exercise 3.3.6 (on page 58): X and Yare independent with 2X

= 9 and2Y= 3. Compute Var(4X 2Y+ 6).

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Definition: The standard deviation of a randomvariable X, X, is computed as X = 2X.

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Families of Discrete Distributions: Some families of

distributions are frequently used and moreimportant. We summarize the a few most usefulones.

Uniform Distribution;

Geometric Distribution;

Bernoulli Distribution;

Binomial Distribution;

Poisson Distribution.

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Uniform Distribution

Definition A random variable X follows a discrete

uniform distribution if the probability ofX takingeach possible value x is equally likely.

Example. Roll a fair dice. Let X= outcome. Xcan take 6 values, andf(X =x) = 16 , where x= 1,

, 6.

Compute E(X) and Var(X) when X can take n values, i.e.x=x1, , xn.

E(X) =

nk=1xkn

;

Var(X) =n

k=1x2k

n n

k=1xkn

2;

F(x) =

#{x1, ,xn}x1n

.

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Geometric Distribution

Definition A random variable X follows a geometric

distribution with parameterp

if its probabilitydensity function f is given by

f(x) = (1 p)x1p, 0

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Bernoulli Distribution

Definition A random variable X follows a Bernoulli

distribution with parameter p if it has probability pto be successful (taking value 1), and probability(1 p) fails (taking value 0).Example. Flip a coin. If a head occurs, let X=1; otherwise, X=0.

With parameter p,

E(X) =p;

Var(X) =p(1 p);

F(x) =

0, ifx

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Binomial Distribution

Definition A random variable X follows a binomialdistribution with parameters n and pif its densityfunction f is given by

f(x) = n

xpx(1 p)nx,

x= 0, 1, 2, , n, 0

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Binomial Distribution (continued)

With parameter n and p, the binomial distribution has

E(X) =np;

Var(X) =np(1 p);F(t) =

[t]x=0

n

x

px(1 p)nx.

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Bi i l Di t ib ti ( ti d)

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Binomial Distribution (continued)

Example 3.5.1 (on page 67) In a study on air traffic controllers, let Xdenote the number of radar signals correctly identified in a 30-minutetime span in which 10 signals arrive. The probability of correctlyidentifying a signal that arrives at random is 0.5.

What is the density function ofX?

f(x) =

10x

0.5x(1 0.5)10x, where x= 0, , 10.

Calculate the mean and standard deviation ofX.

E(X

) = 10 0.5 = 5, X = 2

X =

10 0.5 0.5 1.58.

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Binomial Distribution (continued)What is the probability that at most 7 signals will be identifiedcorrectly?

F(7) = 1 f(10) f(9) f(8)= 1

1010

0.510(1 0.5)1010

10

9

0.59(1 0.5)109

108

0.58(1 0.5)108

0.945What is the probability that 2 X 7?

F(7) F(1) = F(7) f(0) f(1) 0.945

10

0

0.50(1 0.5)100

101

0.51(1 0.5)101

0.935

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Poisson Distribution

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Poisson Distribution

Definition A random variable Xfollows a poisson

distribution with parameter if its density functionf is given by

f(x) =ex

x!

.

Theorem For any real number z, we have

ez = 1 +z+ z2

2!+ z

3

3!+ z

4

4!+ .

This is the Maclaurin series for ez.

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Poisson Distribution (continued)

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Poisson Distribution (continued)

With parameter , a poisson distribution has

E(X) =;

Var(X) =;

F(t) =[t]

x=0f(x) =[t]

x=0ex

x! .

A random variable follows a poisson distribution in manywaiting-for-occurence applications. Generally speaking, X follows apoisson distribution in following cases.

Xdenotes the number of car accidents in a month.

Xdenotes the number of customers coming for service in 5 minutes.Xdenotes the number of incoming calls in a period of time.

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Poisson Distribution (continued)

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Poisson Distribution (continued)Example. Consider a telephone operator who, on the average, handles 5calls every 3 minutes. Model the number of calls in a minute by apoisson distribution. What is the probability that there will be no calls in

the next minute? At least two calls?

Solution:Define Xas the number of calls in a minute. Then E(X) = 53 . So

P(no calls in the next minute) = P(X = 0)

= e5/3(5/3)0

0!

= e5/3 = 0.189;

P(at least two calls in the next minute) = P(X 2)= 1 f(0) f(1)

= 1 0.189 e5/3(5/3)1

1!= 0.496.

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Definition: Let Xbe a random variable. The kth

ordinary moment for X is defined as E(Xk).

So, by definition, the mean is the first ordinary moment. E(Xk) helps todescribe distribution characters ofX.

Definition: The moment generating function (MGF)

for a random variable X is denoted by mX(t), and isgiven by

mX(t) = E(etX),

provided this expectation is finite for all realnumbers tin some open interval (h, h).Why MGF? Because MGF helps to find E(Xk) for any k.

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Example: Random variable X follows a discrete

uniform distribution with x=x1, x2, , xn. Findthe moment generating function for X.

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Example: Random variable X follows a Bernoulli

distribution with parameter p. Find the momentgenerating function for X.

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Example: Random variable X follows a binomial

distribution with parameters n and p. Find themoment generating function for X.

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Example: Random variable X follows a geometric

distribution with parameter p. Find the momentgenerating function for X.

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Example: Random variable Xfollows a poisson

distribution with parameter . Find the momentgenerating function for X.

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