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

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

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

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) ? Y 亦為一隨機變數 Y = g(X) 隨機變數之函式 亦為隨機變數。

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) ? Y 亦為一隨機變數 Y = g(X) 若 p X (x) 已知， p Y (y) =?

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

Example 17 這瓶十元 這瓶只要五元 福氣啦 !!!

Example 17 這瓶十元 這瓶只要五元 福氣啦 !!!

Example 17 這瓶十元 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.

Example 18 n=10, p=0.2. Pay 100$, #bottles ( X 3 ) obtained?

Example 18 n=10, p=0.2. Pay 100$, #bottles ( X 3 ) obtained? Let Y (  X 3 ) denote #lucky bottles obtained.

Discrete Random Vectors Chapter 3-2 Discrete Random Variables

Definition  Random Vectors A discrete r -dimensional random vector X is a function X:   R r with a finite or countable infinite image of {x 1, x 2, …}.

Example 19

11

22

Definition  Joint Pmf Let random vector X = (X 1, X 2, …, X r ). The joint pmf (jpmf) for X is defined as p X (x) = P(X 1 = x 1, X 2 = x 2, …, X r = x r ), where x = (x 1, x 2, …, x r ).

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 1. p(x)  0, x  R r ; 2. {x | p(x)  0} is a finite or countably infinite subset of R r ; 3.

Definition  Marginal Probability Mass Functions Let X = (X 1, …, X i, …, X r ) be an r -dimensional random vectors. The i th marginal probability mass function defined by

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

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

Example 22 4 X = # Y = # 1. p X,Y (x, y) = ? 2. p X (x) = ? p Y (y) = ? 3. p(X < 3)= ? 4. p(X + Y < 4)= ?

p X,Y (x, y) Example 22 4 X = # Y = # 1. p X,Y (x, y) = ? 2. p X (x) = ? p Y (y) = ? 3. p(X < 3)= ? 4. p(X + Y < 4)= ?

p X,Y (x, y) Example 22 4 X = # Y = # 1. p X,Y (x, y) = ? 2. p X (x) = ? p Y (y) = ? 3. p(X < 3)= ? 4. p(X + Y < 4)= ?

p X,Y (x, y) Example 22 4 X = # Y = # 1. p X,Y (x, y) = ? 2. p X (x) = ? p Y (y) = ? 3. p(X < 3)= ? 4. p(X + Y < 4)= ?

p X,Y (x, y) Example 22 4 X = # Y = # 1. p X,Y (x, y) = ? 2. p X (x) = ? p Y (y) = ? 3. p(X < 3)= ? 4. p(X + Y < 4)= ?

p X,Y (x, y) Example 22 4 X = # Y = # 1. p X,Y (x, y) = ? 2. p X (x) = ? p Y (y) = ? 3. p(X < 3)= ? 4. p(X + Y < 4)= ?

Independent Random Variables Chapter 3-2 Discrete Random Variables

Definition Let X 1, X 2, …, X r be r discrete random variables having densities, respectively. These random variables are said to be mutually independent if their jpdf p(x 1, x 2, …, x r ) satisfies

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

Example 23

Fact  ? ? ?



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

Example 24

Z 1 有何意義 ?

Example 24

p’p’ p’p’

Fact: cdf pmf

Example 24

Multinomial Distributions Chapter 3-2 Discrete Random Variables

Generalized Bernoulli Trials A sequence of n independent trials. Each trial has r distinct outcomes with probabilities p 1, p 2, …, p r such that

Multinomial Distributions Define X=(X 1, X 2, …, X r ) st X i is the number of trials that resulted in the i th outcome. satisfies

Multinomial Distributions Define X=(X 1, X 2, …, X r ) st X i is the number of trials that resulted in the i th 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: 1.7 or 11 2.match 3.others X 1  #7 or 11; X 2  #matches; X 3  #others.

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

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). 0 n z  nz  n z 0 n z  nz  n 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}

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 Probabilities Probabilities 機率母函數

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

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

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

Probability Generating Functions pgf 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). 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 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). 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 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). 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 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). 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 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). 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 and Let Z=X+Y. Pf)

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

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

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

Theorem 3

熟記 !!! 請靈活的將它們用於解題

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

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

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

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

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

Example 30 Let X~B(n, p 1 ), Y~B(m, p 2 ) be two independent random variables. U = X + Y V = X  Y Let X~B(n, p 1 ), Y~B(m, p 2 ) be two independent random variables. U = X + Y V = X  Y LetFind p U,V (u, v).

Example 30 Let X~B(n, p 1 ), Y~B(m, p 2 ) be two independent random variables. U = X + Y V = X  Y Let X~B(n, p 1 ), Y~B(m, p 2 ) be two independent random variables. U = X + Y V = X  Y LetFind p U,V (u, v). and