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

2. 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

3. 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

4. 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.

5. 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.

6. 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.

7. 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.

8. 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, …}

9. 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.

10. 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)

11. 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-)

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

13. 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)

14. 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

15. 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}

16. 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)

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

18. 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 .

19. 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?

20. 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.

21. 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?

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

23. 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

24. 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

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

26. 4.5 Variances and moments of discrete random variables
Def Variance of X
Standard deviation of X

27. 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

28. 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=( )/6=7/2
EX2=( )/6=91/6
Var(X)=91/6-(7/2)2=35/12

29. 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)

30. 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

31. 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)

32. 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

33. 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