Handout #5. Discrete Random Variables and Probability Distributions

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1 Hadout #5 Title: Foudatios of Ecoometrics Course: Eco 367 Fall/015 Istructor: Dr. I-Mig Chiu Discrete Radom Variables ad Probability Distributios Radom Variable (RV) Cosider the followig experimet: Toss two fair cois simultaeously. What s the sample space? S (sample space) = HH, HT, TH, TT Suppose we are iterested i the umber of heads i this experimet, the collectio of evets (recall a evet is a subset) would be TT, HT, TH, HH. Let s defie a fuctio that maps each evet to real umbers (o-egative itegers 0, 1, ). The above statemet defies a radom variable, (# of Heads). The ability distributio fuctio will be TT (1/4) HT, TH (1/) HH (1/4) 0 (1/4) 1 (1/) (1/4) Reserve the ability attributes. e.g. = retur of owig a share of stock; it s a radom variable because it s ulikely to kow the outcome whe it s sold (i.e., sellig price > = < buyig price retur > = < 0) e.g. Y = the votig decisio; it s a radom variable because it s ot kow which cadidate the voter is goig to cast a vote for. e.g. Z = whether a idividual is i the labor force; it s a radom variable before the survey is coducted, we have o idea about someoe s employmet status. Which of the above radom variable is discrete? Cotiuous? 1

2 Probability Mass Fuctio (PMF) Def: Let be a discrete radom variable. The ability mass fuctio (pmf) of is the fuctio f : R R defied by f(x) = P( = x) PMF RV(# of Heads) Property of Probability Distributio: a) 0 P( i ) 1 e.g. P ( = 0) = P( = ) = 1/4, P( = 1) = 1/ b) P (i) = 1 e.g. ¼ + ½ + ¼ = 1 Cumulative Distributio Fuctio (CDF) Def: For each y R, let L(y) = { x (S): x y} deote the value of that are less tha or equal to y. The F(y) = P( y) = P( L(y)) = P ( x) = f ( x) x L ( y) x L( y) CDF RV(# of Heads)

3 I order to describe the properties of a ability distributio, a measure called the momets of a distributio is commoly used. They iclude mea, variace, skewess, kurtosis, ad higher momets. Here we oly focus o the first two momets. Mea: The expected value of a radom variable Mea ( ) = E() = * f( = 0* + 1* + * = i i ) Variace: The dispersio of a ability distributio Variace ( ) = E( - ) = ( ) * f( = (0-1) * + (1-1) * + (-1) * = 4 4 i i ) Notice: Variace ( ) = E( - ) = E( ) (E()) = E( ) - Stadard Deviatio ( ) = ( ) * f( = i i ) e.g.: is a radom variable that deotes the umber of heads appeared i a experimet of tossig three fair cois. What is the mea, variace ad stadard deviatio of radom variable? Expected Value of a Fuctio Suppose a fuctio h() is derived from the radom variable, the E(h()) = h ( ) * f ( ) e.g. 3. (pp. 116) A computer store has purchased three computers at $500 apiece. It will sell them for $1000 apiece. The maufacturer has agreed to repurchase ay computers still usold after a specified period at $00 apiece. Let deote the umber of computers sold, ad suppose that p(0) = 0.1, p(1) = 0., p() = 0.3, ad p(3) = 0.4. With h() deotig the profit associated with sellig uits, the give iformatio implies that h() = reveue cost = 1000* + 00*(3 ) What is the expected profit? Theorem: E(a + b*) = a + b*e(), Var(a + b*) = b *Var() 1 3

4 Importat Discrete Radom Variables ad their Distributios (a) Beroulli Distributio Biary radom variables (e.g., healthy/diseased) are abudat i scietific studies. There are also umerous biary radom variables exist i ecoomic studies; e.g., whether a labor is employed or uemployed, whether a worker is a female or male, whether the ecoomy is expadig or cotractig (i.e., busiess cycle), etc. The biary radom variable with possible values 0 ad 1 has a Beroulli distributio with parameter p. Here, P( = 1) = p ad P( = 0) = 1 - p e.g. P( = x) = 0. for x = 0 P( = x) = 0.8 for x = 1 We deote this as Beroulli(p), where 0 p 1. E() = p Var() =p(1 - p) Beroulli Dist

5 (b) Biomial Distributio A sequece of biary radom variables 1,,, is called Beroulli trials if they all have the same Beroulli distributio ad are idepedet. The radom variable Y represetig the umber of times the outcome of iterest occurs i Beroulli trials (i.e., the sum of Beroulli trials) has a Biomial(, p) distributio. I other words, Y = The pmf of a Biomial(, p) specifies the ability of each possible value (itegers from 0 through ) of the radom variable. The pmf is show as follows: b(x;, p) = xp *( x) 1-p, x = 0, 1,,, x E(Y) = p Var(Y) = p(1 - p) I the above, the theoretical (populatio) mea of a radom variable Y with Biomial(, p) distributio is = p. The theoretical (populatio) variace of Y is = p(1 - p). Biomial Distributio (PMF) Y Biomial Distributio (CDF) Y 5

6 e.g (pp. 13~33) Suppose that 0% of all copies of a particular textbook fail a bidig stregth test. Let deote the umber amog 15 radomly selected copies that fail the test. The has a biomial distributio with = 15 ad p = 0.. a) The ability that at most 8 fail the test is. b) The ability that exactly 8 fail is. c) The ability that at least 8 fail is. d) Fially, the ability that betwee 4 ad 7, iclusive, fail is. (c) Poisso distributio A discrete radom variable is said to have a Poisso distributio with parameter > 0, if for x = 0, 1,,... the ability mass fuctio of is give by: P ( = x) = e x! x (e: expoetial fuctio, x!: x factorial) E() = Var() = λ (both the mea ad variace of the Poisso radom variable equal ) Notice that P( = x)/p( = x-1) = x, so we ca obtai the P() recursively startig from P( = 0) = e - Poisso Dist (lambda = ) Poisso Dist (lambda = 5)

7 Poisso Dist (CDF; lambda = ) e.g (pp. 146) Let deote the umber of creatures of a particular type captured i a trap durig a give time period. Suppose that has a Poisso distributio with = 4.5, so o average traps will cotai 4.5 creatures. [The article Dispersal Dyamics of the Bivalve Gemma gemma i a Patchy Eviromet (Ecol. Moogr., 1995: 1 0) suggests this model; the bivalve Gemma gemma is a small clam.] The ability that a trap cotais exactly five creatures is. The ability that a trap has at most five creatures is. Theorem: Suppose that i the biomial pmf b(x;, p) we let goes to ifiity ( ) ad p is close to 0 (p 0) i such a way that *p approaches a value > 0. The the Biomial distributio ca be approximated by a Poisso distributio. 7

Discrete Probability Functions

Discrete Probability Functions Discrete Probability Fuctios Daiel B. Rowe, Ph.D. Professor Departmet of Mathematics, Statistics, ad Computer Sciece Copyright 017 by 1 Outlie Discrete RVs, PMFs, CDFs Discrete Expectatios Discrete Momets