Chapter 3-2 Discrete Random Variables 主講人:虞台文

Content Functions of a Single Discrete Random Variable Discrete Random Vectors Independent of Random Variables Multinomial Distributions Sums of Independent Variables  Generating Functions Functions of Multiple Random Variables

Chapter 3-2 Discrete Random Variables Functions of a Single Discrete Random Variable

計程車司機的心聲 這傢伙上車後會要跑幾公里(X)? X為一隨機變數

這傢伙上車後我可以從他口袋掏多少錢(Y)? 隨機變數之函式亦為隨機變數。 Y = g(X) 計程車司機的心聲  這傢伙上車後會要跑幾公里(X)? X為一隨機變數 Y亦為一隨機變數 這傢伙上車後我可以從他口袋掏多少錢(Y)?

這傢伙上車後我可以從他口袋掏多少錢(Y)? Y = g(X) 若pX(x)已知， pY(y)=? 計程車司機的心聲  這傢伙上車後會要跑幾公里(X)? X為一隨機變數 Y亦為一隨機變數 這傢伙上車後我可以從他口袋掏多少錢(Y)?

The Problem Y = g(X) and pX(x) is available.

這瓶十元 Example 17 這瓶只要五元 福氣啦!!!

這瓶十元 Example 17 這瓶只要五元 福氣啦!!!

這瓶十元 Example 17 這瓶只要五元 福氣啦!!!

Example 17

Example 18 n=10, p=0.2.

Example 18 n=10, p=0.2.

Example 18 n=10, p=0.2.

Pay 100$, #bottles (X3) obtained? Example 18 n=10, p=0.2.

Example 18 n=10, p=0.2. Pay 100$, #bottles (X3) obtained? Let Y (X3) denote #lucky bottles obtained.

Chapter 3-2 Discrete Random Variables Discrete Random Vectors

Definition  Random Vectors A discrete r-dimensional random vector X is a function X:   Rr with a finite or countable infinite image of {x1, x2, …}.

Example 19

1 Example 19

2 Example 19

pX(x) = P(X1 = x1, X2 = x2, … , Xr = xr), Definition  Joint Pmf Let random vector X = (X1, X2, …, Xr). The joint pmf (jpmf) for X is defined as pX(x) = P(X1 = x1, X2 = x2, … , Xr = xr), where x = (x1, x2, … , xr).

Example 20 There are three cards numbered 1, 2 and 3. Randomly draw two cards among them without replacement. Let X, Y represent the number of the 1st and 2nd card, respectively. Find the jpmf of X, Y. X Y

Example 20 There are three cards numbered 1, 2 and 3. Randomly draw two cards among them without replacement. Let X, Y represent the number of the 1st and 2nd card, respectively. Find the jpmf of X, Y. X Y

Properties of Jpmf's p(x)  0, x  Rr; {x | p(x)  0} is a finite or countably infinite subset of Rr;

Definition  Marginal Probability Mass Functions Let X = (X1, …, Xi , …, Xr) be an r-dimensional random vectors. The ith marginal probability mass function defined by

Example 21 Find pX(x) and pY (y) of Example 20. X Y

Example 21 Find pX(x) and pY (y) of Example 20. X Y

Example 22 X = # 4 Y = # pX,Y(x, y) = ? pX (x) = ? pY (y) = ?

Example 22 X = # 4 Y = # pX,Y(x, y) pX,Y(x, y) = ? pX (x) = ? pY (y) = ? p(X < 3)= ? p(X + Y < 4)= ? 4 Example 22 pX,Y(x, y)

Example 22 X = # 4 Y = # pX,Y(x, y) pX,Y(x, y) = ? pX (x) = ? pY (y) = ? p(X < 3)= ? p(X + Y < 4)= ? 4 Example 22 pX,Y(x, y)

Example 22 X = # 4 Y = # pX,Y(x, y) pX,Y(x, y) = ? pX (x) = ? pY (y) = ? p(X < 3)= ? p(X + Y < 4)= ? 4 Example 22 pX,Y(x, y)

Example 22 X = # 4 Y = # pX,Y(x, y) pX,Y(x, y) = ? pX (x) = ? pY (y) = ? p(X < 3)= ? p(X + Y < 4)= ? 4 Example 22 pX,Y(x, y)

Example 22 X = # 4 Y = # pX,Y(x, y) pX,Y(x, y) = ? pX (x) = ? pY (y) = ? p(X < 3)= ? p(X + Y < 4)= ? 4 Example 22 pX,Y(x, y)

Chapter 3-2 Discrete Random Variables Independent Random Variables

Definition Let X1, X2, …, Xr be r discrete random variables having densities , respectively. These random variables are said to be mutually independent if their jpdf p(x1, x2, …, xr) satisfies

Example 23 Tossing two dice, let X, Y represent the face values of the 1st and 2nd dice, respectively. 1. pX,Y (x, y) = ?. 2. Are X, Y independent?

Example 23

Fact  ? ? ?

Fact

Fact 

Example 24 Consider Example 23. Find P(X  2, Y  4).

Example 24

Example 24

Example 24 Z1有何意義?

Example 24

Example 24

Example 24

Example 24 p’ p’

Example 24

Example 24 Fact: cdf pmf

Example 24

Example 24

Example 24

Chapter 3-2 Discrete Random Variables Multinomial Distributions

Generalized Bernoulli Trials A sequence of n independent trials. Each trial has r distinct outcomes with probabilities p1, p2, …, pr such that

Multinomial Distributions Define X=(X1, X2, …, Xr) st Xi is the number of trials that resulted in the ith outcome. satisfies

Multinomial Distributions Define X=(X1, X2, …, Xr) st Xi is the number of trials that resulted in the ith outcome. satisfies

Example 26 If a pair of dice are tossed 6 times, what is the probability of obtaining a total of 7 or 11 twice, a matching pair one, and any other combination 3 times? Three outcomes: 7 or 11 match others X1  #7 or 11; X2  #matches; X3  #others.

Chapter 3-2 Discrete Random Variables Sums of Independent Variables  Generating Functions

The Sum of Independent Random Variables

Example 27 Let X, Y be two independent random variables each uniformly distributed over 0, 1, 2, …, n. Find P(X+Y = z).

Example 27 Let X, Y be two independent random variables each uniformly distributed over 0, 1, 2, …, n. Find P(X+Y = z). Case 1: z{0, 1, …, n} Case 2: z{n+1, n+2, …, 2n} n n z  n z z  n z

Example 27 Let X, Y be two independent random variables each uniformly distributed over 0, 1, 2, …, n. Find P(X+Y = z). Case 1: z{0, 1, …, n} Case 2: z{n+1, n+2, …, 2n}

Example 27 Let X, Y be two independent random variables each uniformly distributed over 0, 1, 2, …, n. Find P(X+Y = z). Case 1: z{0, 1, …, n} Case 2: z{n+1, n+2, …, 2n}

Probability Generating Functions 機率母函數 Probability Generating Functions Probabilities Probabilities

Probability Generating Functions pgf Probability Generating Functions Let X be a nonnegative integer-valued random variable. Its probability generating function GX(t) is defined as:

Probability Generating Functions pgf Probability Generating Functions Let X be a nonnegative integer-valued random variable. Its probability generating function GX(t) is defined as: x 2 1

Probability Generating Functions pgf Probability Generating Functions Let X be a nonnegative integer-valued random variable. Its probability generating function GX(t) is defined as: x 2 1

Probability Generating Functions pgf Probability Generating Functions Compute the pgf’s for the following distributions: 1. X ~ B(n, p); 2. Y ~ P(); 3. Z ~ G(p); 4. U ~ NB(r, p).

Probability Generating Functions pgf Probability Generating Functions Compute the pgf’s for the following distributions: 1. X ~ B(n, p); 2. Y ~ P(); 3. Z ~ G(p); 4. U ~ NB(r, p).

Probability Generating Functions pgf Probability Generating Functions Compute the pgf’s for the following distributions: 1. X ~ B(n, p); 2. Y ~ P(); 3. Z ~ G(p); 4. U ~ NB(r, p).

Probability Generating Functions pgf Probability Generating Functions Compute the pgf’s for the following distributions: 1. X ~ B(n, p); 2. Y ~ P(); 3. Z ~ G(p); 4. U ~ NB(r, p).

Probability Generating Functions pgf Probability Generating Functions Compute the pgf’s for the following distributions: 1. X ~ B(n, p); 2. Y ~ P(); 3. Z ~ G(p); 4. U ~ NB(r, p). Exercise

Important Generating Functions

Theorem 2  Sums of Independent Random Variables Let X, Y be two independent, nonnegative integer-valued random variables. Then,

Theorem 2  Sums of Independent Random Variables Pf) Let Z=X+Y.

Theorem 2  Sums of Independent Random Variables Fact: and . . .

Example 29 Use pgf to recompute Example 27. Example 27 Let X, Y be two independent random variables each uniformly distributed over 0, 1, 2, …, n. Find P(X+Y = z). Example 29 Use pgf to recompute Example 27.

Example 29 Use pgf to recompute Example 27. Example 27 Let X, Y be two independent random variables each uniformly distributed over 0, 1, 2, …, n. Find P(X+Y = z). Example 29 Use pgf to recompute Example 27.

Theorem 3

Theorem 3 表何意義?

Theorem 3

Theorem 3 表何意義?

Theorem 3

Theorem 3 表何意義?

Theorem 3

Theorem 3 表何意義?

Theorem 3

Theorem 3 表何意義?

Theorem 3

熟記!!!請靈活的將它們用於解題 Theorem 3

Chapter 3-2 Discrete Random Variables Functions of Multiple Random Variables

Functions of Multiple Random Variables Let X, Y be two random variables with jpmf pX,Y(x, y). 1-1 Suppose that pU,V(u, v)=?

Functions of Multiple Random Variables Let X, Y be two random variables with jpmf pX,Y(x, y). 1-1 Suppose that pU,V(u, v)=? Example: pX,Y(x, y)  已知 pU,V(u, v) = ? X $/month Y $/month

Functions of Multiple Random Variables 1-1 implies invertible. Functions of Multiple Random Variables Let X, Y be two random variables with jpmf pX,Y(x, y). 1-1 Suppose that pU,V(u, v)=? Example: pX,Y(x, y)  已知 pU,V(u, v) = ?

Functions of Multiple Random Variables 1-1 implies invertible. Functions of Multiple Random Variables Let X, Y be two random variables with jpmf pX,Y(x, y). 1-1 Suppose that pU,V(u, v)=?

Example 30 Let X~B(n, p1), Y~B(m, p2) be two independent random variables. U = X + Y V = X  Y Let Find pU,V(u, v).

Example 30 Let X~B(n, p1), Y~B(m, p2) be two independent random variables. U = X + Y V = X  Y Let Find pU,V(u, v). and