Special random variables Chapter 5 Some discrete or continuous probability distributions

Some special random variables Bernoulli Binomial Poisson Hypergeometric Uniform Normal and its derivatives Chi-square T-distribution F-distribution Exponential Gamma

Bernoulli random variable A random variable X is said to be a Bernoulli random variable if its probability mass function is given as the following: P{X=0}=1-p, P{X=1}=p; (you may assume X=1 when the experimental outcome is successful and X=0 when it is failed.) E[X]=1×P{X=1}+0×P{X=0}=p Var[X]=E[X2]-E[X]2 =p(1-p)

Binomial random variable Suppose there are n independent Bernoulli trials, each of which results in a “ success ” with the probability p. If X represents the number of successes that occur in the n trials, then X is said to be a binomial random variable with parameters (n,p). P{X=i}= n!/[(n-i)! ×i!] × pn(1-p)(n-i), i=0, 1, 2, … n

The expectation of binomial random variable The Binomial random X is composed of n independent Bernoulli trials ∴ X=Σ 1~n x i, x i =1 when the i th trial is a success or x i =0 otherwise E[X]=Σ 1~n E[x i ]=n×p Var[X]=Σ 1~n Var[x i ]=np(1-p)

Patterns of binomial distribution If p=0.5, then X will distribute symmetrically If p>0.5, then X will be a left-skewed distribution If p<0.5, then X will be a right-skewed distribution If n  ∞, then X will distribute as a symmetric bell/normal pattern

Poisson random variable The random variable X is a Poisson distribution if its prob. mass function is given by (By the Taylor series)

Expectation and variance of Poisson distribution By using the moment generating function

Poisson vs. Binomial A binomial distribution, (n,p),  a Poisson with meanλ=np, when n is large and p is small. In other words, if the successful probability of trial is very small, then the accumulative many trials distribute as a Poisson. The probability of one person living over 100 years of age is 0.01, then the mean number of over-100-year-old persons 10 may occur within 1000 elders. λ means the average occurrence among a large number of experiments, in contrast to, p means the occurring chance of every trial

Comparisons A binomial random variable with p near to 0.5 A Poisson random variable approximates to a binomial distribution when n becomes large.

Hypergeometric random variable There are N+M objects, of which N are desirable and the other M are defective. A sample size n is randomly chosen from N+M without replacements. Let X be the number of desirable within n chosen objects. Its probability mass function as the following. We said X is a hypergeometric distribution with parameters (N,M,n)

Expectation & variance of hypergeometric distribution

Expectation & variance of hypergeometric distribution (cont.) Moreover, if N+M increases to ∞, then Var(X) converges to np(1-p), which is the variance of a binomial random variable with parameters (n,p).

hypergeometric vs. binomial Let X, and Y be independent binomial random variables having respective parameters (n,p) and (m,p). The conditional p.m.f. of X given that X+Y=k is a hypergeometric distribution with parameters (n,m,k).

The Uniform random variable A random variable X is said to be uniformly distributed over the interval [α,β] if its p.d.f. is given by

Pattern of uniform distribution

Expectation and variance of uniform distribution E[X]=∫ α~ β x [1/(β-α)]d x =(α+β)/2 Var(X)=E[X2]-E[X]2 =(β-α)2/12 P.161, Example 5.4b

Normal random variable A continuous r.v X has a normal distribution with parameter μ and σ2 if its probability density function is given by: We write X~N(μ,σ2) By using the M.G.F., we obtain E[X]=ψ ’ (0)=μ, and Var[X]=ψ ” (0)-ψ ’ (0)2=σ2

Standard normal distribution The Cumulative Distribution function of Z

Percentiles of the normal distribution Suppose that a test score distributes as a normal distribution with mean 500 and standard deviation of 100. What is the 75th percentile score of this test?

Characteristics of Normal distribution P{ ∣ X-μ ∣ <σ}=ψ(-1<X-μ/σ<1)=ψ(-1<Z<1)= ψ(1)-ψ(-1)=2ψ(1) -1=2× P{ ∣ X-μ ∣ <2σ}=ψ(-2<X-μ/σ<2)=ψ(-2<Z<2)= ψ(2)-ψ(-2)=2ψ(2) -1=2× P{ ∣ X-μ ∣ <3σ}=ψ(-3<X-μ/σ<3)=ψ(-3<Z<3)= ψ(3)-ψ(-3)=2ψ(3) -1=2×

The pattern of normal distribution

Exponential random variables A nonnegative random variable X with a parameter λ obeying the following pdf and cdf is called an exponential distribution. The exponential distribution is often used to describe the distribution of the amount of time until some specific event occurs. The amount of time until an earthquake occurs The amount of time until a telephone call you receive turns to be the wrong number

Pattern of exponential distribution

Expectation and variance of exponential distribution E[X]=ψ ’ (0)=1/λ E[X] means the average cycle time, λ presents the occurring frequency per time interval Var(X)=ψ ” (0)-ψ ’ (0)2=1/λ2 The memoryless property of X P{X>s+t/X>t}=P{X>s}, if s, t ≧ 0; P{X>s+t}=P{X>s} ×P{X>t}

Poisson vs. exponential Suppose that independent events are occurring at random time points, and let N(t) denote the number of event that occurs in the time interval [0,t]. These events are said to constitute a Poisson process having rate λ, λ>0, if N(0)=0; The distribution of number of events occurring within an interval depends on the time length and not on the time point. lim h  0 P{N(h)=1}/h=λ lim h  0 P{N(h) ≧ 2}/h=0

Poisson vs. exponential (cont.) P{N(t)=k}=P{k of the n subintervals contain exactly 1 event and the other n-k contain 0 events} P{ecactly 1 event in a subinterval t/n} ≒ λ(t/n) P{0 events in a subinterval t/n} ≒ 1-λ(t/n) P{N(t)=k} ≒, a binomial distribution with p=λ(t/n) A binomial distribution approximates to a Poisson distribution with k=n(λt/n) when n is large and p is small.

Poisson vs. exponential (cont.) Let X 1 is the time of first event. P{X 1 >t }=P{N(t)=0} (0 events in the first t time length)=exp(-λt) ∵ F(t)=P{X 1 ≦ t }=1- exp(-λt) with mean 1/λ ∴ X 1 is an exponential random variable Let X n is the time elapsed between (n-1)st and nth event. P{X n >t/ X n-1 =s}=P{N(t)=0} =exp(-λt) ∵ F(t)=P{X n ≦ t }=1- exp(-λt) with mean 1/λ ∴ X n is also an exponential random variable

Gamma distribution See the gamma definition and proof in p If X 1 and X 2 are independent gamma random variables having respective parameters (α 1,λ) and (α 2,λ), then X 1 +X 2 is a gamma random variable with (α 1 +α 2,λ) The gamma distribution with (1,λ) reduces to the exponential with the rate λ. If X 1, X 2, … X n are independent exponential random variables, each having rate λ, then X 1 + X 2 + … +X n is a gamma random variable with parameters (n,λ)

Expectation and variance of Gamma distribution The gamma distribution with (α,1) reduces to the normal distribution when α becomes large. The patters of gamma distribution move from right-skewed toward symmetric as α increases. See p.183, using the G.M.F. to compute E[X]=α/λ Var(X)=α/(λ2 )

Patterns of gamma distribution

Derivatives from the normal distribution The chi-square distribution The t distribution The F distribution

The Chi-square distribution If Z 1, Z 2, … Z n are independent standard normal random variables, the X, defined by X=Z12+Z22+ … Zn2, is called chi-square distribution with n degrees of freedom, and denoted by X~Xn2 If X 1 and X 2 are independent chi-square random variables with n 1 and n 2 degrees of freedom, respectively, then X 1 +X 2 is chi-square with n 1 +n 2 degrees of freedom.

p.d.f. of Chi-square The probability density function for the distribution with r degrees of freedom is given by

The relation between chi-square and gamma A chi-square random variable with n degrees of freedom is identical to a gamma random variable with (n/2,1/2), i.e., α=n/2,λ=1/2 The expectation of chi-square distribution E[X] is the same as the expectation of gamma withα=n/2,λ=1/2, so E[X]=α/λ=n (degrees of freedom) The variance of chi-square distribution Var(X)=α/(λ 2 )=2n The chi-square distribution moves from right- skewed toward symmetric as the degree of freedom n increases.

Patterns of chi-square distribution

The t distribution If Z and Xn2 are independent random variables, with Z having a std. normal dist. And Xn2 having a chi-square dist. with n degrees of freedom, then T n defined by For large n, the t distribution approximates to the standard normal distribution The t distribution move from flatter and having thicker tails toward steeper and having thinner tails as n increases.

p.d.f. of t-distribution B(p,q) is the beta functionbeta function

Patterns of t distribution

The F distribution If Xn2 and Xm2 are independent chi-square random variables with n and m degrees of freedom, respectively, then the random variable F n,m defined by F n,m is said an F-distribution with n and m degrees of freedom F 1,m is the same as the square of t-distribution, ( T m ) ²

p.d.f of F-distribution where is the gamma function, B(p,q) is the beta function,gamma functionbeta function

Patterns of F-distribution

Relationship between different random variables Gamma(α, λ) Poisson(λ) Exponential(λ) Normal (μ,σ2) Binomial (n, p)Bernoulli (p) Z (0, 1) n  ∞ p  0 λ=np n  ∞, p  0.5 Repeated trials Standardization Z=(X-μ)/σ The very small time partition α=1 Chi-square (n) t distribution F distribution λ=1, α  ∞ α=n/2 λ=1/2

Homework #4 Problem 5,12,27,31,39,44