Distribution

p10.
Probability of the event in terms of F F(b) – F(a) F(b-) – F(a) F(b) – F(a-) F(b-) –F(a-)
a < X <= b a<X<b a <= X <= b a <= X < b
Distribution functions
X ∈I
p2.
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.
Ex 4.7 The distribution function of a random variable X is given by
x<0 0 x / 4 0 ≤ x <1 F ( x ) = 1 / 2 1≤ x < 2 x / 12 + 1 / 2 2 ≤ x < 3 1 x≥3
p4.
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.
p18.
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, ∞ ∞ k ( 1 )k = EX = 2 1 = 1+1+1+ = ∞
p8.
Distribution functions
Properties of the distribution functions: 1. F is nondecreasing; that is, if t<u, then F(t)<=F(u). 2. lim F (t ) = 1 t →∞ 3. limt → ∞ F (t ) = 0 4. F is right continuous; that is, for every t in R, F(t+)=F(t)
p14.
Discrete random variables
Def The probability 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) ∞ p ( xi ) = 1 i =1
p11.
Distribution functions
Compute the following quantities: (a)P(X<2) (b)P(X=2) (c)P(1<=X<3) (d)P(X>3/2) (e)P(X=5/2) (f)P(2<X<=7)
p12.
Distribution functions
The set of all possible values of X is {1, 2, 3, …}
89 P ( X = i ) = P ( NNN ... N T ) = 90
i 1 i 1
1 90
p7.
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.
E( X ) =
x∈A
∑ xp( x)
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, X or .
1
∞
∑
i =1
(2 / 3)i
1 = 2
p16.
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 function p(x) is defined by
p20.
Expectations of discrete random variables
Sol: Let Y=max Xi; then for k=n, n+1,
p3.
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 rete random variables
Ex 4.12 Can a function of the form
c( 2 / 3) x p( x ) = 0
x = 1,2,3,... elsewhere.
be a probability function? Sol: c =
Ex 4.19 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?
t<0 0 F (t ) = 2(t 10) 10 ≤ t < 10.5 1 t ≥ 10.5
p13.
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}
4. Distribution Functions and Discrete Random Variables
2010/9/2
1
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, {s : X ( s ) ∈ I } is an event. In probability, the set {s : X ( s ) ∈ I } is often abbreviated as { X ∈ I } or simply as
p9.
Distribution functions
Event concerning X X <= a X>a X<a X >= a X=a Probability Event of the event concerning X in terms of F F(a) 1 – F(a) F(a-) 1 – F(a-) F(a) – F(a-)
p17.
Expectations of discrete random variables