Chap 4 Distribution Functions and Discrete Random Variables Ghahramani 3rd edition 2019/1/3

Outline 4.1 Random variables 4.2 Distribution functions 4.3 Expectations of random variables 4.4 Basic Theorems 4.5 Variances and moments of discrete random variables 4.6 Standardized random variables

4.1 Random variables Def A real-valued function X: SR is called a random variable of the experiment if, for each interval I R, is an event. In probability, the set is often abbreviated as or simply as

Random variables Ex 4.1 Suppose that 3 cards are drawn from an ordinary deck of 52 cards, 1-by-1, at random and with replacement. Let X be the number of spades drawn; then X is a random variable.

Random variables If an outcome of spades is denoted by s, and other outcomes are represented by t, then X is a real-valued function defined on the sample space S={(s,s,s), (t,s,s), (s,t,s), (s,s,t), (s,t,t), (t,s,t), (t,t,s), (t,t,t)}, by X(s,s,s)=3, X(s,t,s)=2, X(s,s,t)=2, X(s,t,t)=1, and so on.

Random variables P(X=0)=P({(t,t,t)})=27/64 P(X=1)=P({(s,t,t),(t,s,t),(t,t,s)})=27/64 P(X=2)=P({(s,s,t),(s,t,s),(t,s,s)})=9/64 P(X=3)=P({(s,s,s)})=1/64 If the cards are drawn without replacement, P(X=i)=C(13,i)C(39,3-i)/C(52,3) for i=0,1,2,3.

Random variables Ex 4.3 In the U.S., the number of twin births is approximately 1 in 90. Let X be the number of births in a certain hospital until the first twins are born. X is a random variable.

Random variables Denote twin births by T and single births by N. The X is a real-valued function defined on the sample space The set of all possible values of X is {1, 2, 3, …}

4.2 Distribution functions Def If X is a random variable, then the function F defined on by F(t)=P(X<=t) is called the distribution function of X.

Distribution functions Properties of the distribution functions: F is nondecreasing; that is, if t<u, then F(t)<=F(u). 2. 3. 4. F is right continuous; that is, for every t in R, F(t+)=F(t)

Distribution functions Event concerning X Probability of the event in terms of F X <= a F(a) a < X <= b F(b) – F(a) X > a 1 – F(a) a < X < b F(b-) – F(a) X < a F(a-) a <= X <= b F(b) – F(a-) X >= a 1 – F(a-) a <= X < b F(b-) –F(a-) X = a F(a) – F(a-)

Distribution functions Ex 4.7 The distribution function of a random variable X is given by

Distribution functions Compute the following quantities: P(X<2) P(X=2) P(1<=X<3) P(X>3/2) P(X=5/2) P(2<X<=7)

Distribution functions Ex 4.9 Suppose that a bus arrives at a station every day between 10:00 A.M. and 10:30 A.M., at random. Let X be the arrival time; find the distribution function of X and sketch its graph. Sol

4.3 Discrete random variables Def Whenever the set of possible values that a random variable X can assume is at most countable, X is called discrete. Examples of set measure finite set {0, 1, 2} countable set {1, 2, 3, 4, … } uncountable set {x: x >= 0}

Discrete random variables Def The probability mass function p of a random variable X whose set of possible values is {x1, x2, x3, …} is a function from R to R that satisfies the following properties. (a) p(x)=0 if x {x1, x2, x3, …} (b) p(xi)=P(X=xi) and hence p(xi)>=0 (c)

Discrete random variables Ex 4.12 Can a function of the form be a probability function? Sol:

4.4 Expectations of discrete random variables Def The expected value of a discrete random variable X with the set of possible values A and probability mass function p(x) is defined by We say that E(X) exists if this sum converges absolutely. E(X) is also called the mean or the expectation of X and is also denoted by EX, or .

Expectations of discrete random variables Ex 4.18 (St. Petersburg Paradox) In a game, the player flips a fair coin successively until he gets a heads. If this occurs on the kth flip, the player wins 2k dollars. Question: To play this game, how much should a person, who is willing to play a fair game, pay?

Expectations of discrete random variables Sol: Let X be the amount of money the player wins. Then X is a random variable with the set of possible values {2, 4, 8, …} and P(X=2k)=1/2k, k=1, 2, 3, … Therefore, This shows that the game remains unfair even if a person pays the largest possible amount to play it.

Expectations of discrete random variables Ex 4.20 The tanks of a country’s army are numbered 1 to N. In a war this country loses n random tanks to the enemy, who discovers that the captured tanks are numbered. If X1, X2, …, Xn are the numbers of the captured tanks, what is E(max Xi)? How can the enemy use E(max Xi) to find an estimate of N, the total number of this country’s tanks?

Expectations of discrete random variables Sol: Let Y=max Xi; then for k=n, n+1, n+2, …, N,

Expectations of discrete random variables If enemy captures 12 tanks and the maximum of the numbers of the tanks captured is 117, then we get N is around (13/12)117-1 = 126

Expectations of discrete random variables Thm 4.1 If X is a constant random variable, that is, if P(X=c)=1 for a constant c, then EX=c. Thm 4.2 Let g be a real-valued function. Then g(X) is a random variable with

Expectations of discrete random variables Coro Let g1, g2, …, gn be real-valued functions, and let a1, a2, …, an be real numbers. Then

4.5 Variances and moments of discrete random variables Def Variance of X Standard deviation of X

Variances and moments of discrete random variables Thm 4.3 Var(X) = EX2 – (EX)2 Proof: Var(X) = E[(X-EX)2] = E[X2 – 2XEX + (EX)2] = E(X2) – 2EXEX +(EX)2 = E(X2) – (EX)2 Application: (EX)2 <= EX2

Variances and moments of discrete random variables Ex 4.27 What is the variance of the random variable X, the outcome of rolling a fair die? Sol: EX=(1+2+3+4+5+6)/6=7/2 EX2=(1+4+9+16+25+36)/6=91/6 Var(X)=91/6-(7/2)2=35/12

Variances and moments of discrete random variables Thm 4.4 Var(X)=0 iff X is constant with probability 1. Thm 4.5 Var(aX+b)=a2Var(X)

Variances and moments of discrete random variables Ex 4.28 EX=2 and E[X(X-4)]=5. Var(–4X+12)=? Sol: E[X2-4X]=EX2 –4EX=5 so EX2=5+4x2=13 Hence Var(X)=EX2 –(EX)2 =13-22=9 By Thm 4.5 Var(–4X+12)=16x9=144

Variances and moments of discrete random variables Def Let w be a given point. X is more concentrated about w than is Y. If for all t > 0 P(|Y-w|<=t) <= P(|X-w|<=t) Thm 4.6 Suppose that EX=EY=a. If X is more concentrated about a than is Y, then Var(X)<=Var(Y)

Variances and moments of discrete random variables Def Let c be a constant, n>=0 be an integer, and r>0 be any real number. E[g(X)] Definition E(Xn) E(|X|r) E(X-c) E[(X-c)n] E[(X-EX)n] The nth moment of X The rth absolute moment of X The 1st moment of X about c The nth moment of X about c The nth central moment of X

4.6 Standardized random variables Def Let X be a random variable with mean and standard deviation . The random variable is called the standardized X. We have