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Discrete Random Variables and Probability Distributions

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Discrete Random Variables and Probability Distributions
Random Variables and Probability Distributions

 A random variable is a function that assigns a numerical value to each sample point in
a sample space.  A random variable reflects the aspect of a random experiment that is of interest to us.  There are two types of random variables 1. Discrete random variable 2. Continuous random variable A random variable is discrete if it can assume only a countable number of values. A random variable is continuous if it can assume an uncountable number of values. Discrete Probability Distribution

 A table, formula, or graph that lists all possible values a discrete random variable can
assume, together with associated probabilities, is called a discrete probability distribution  To calculate P( X  x) , the probability that the random variable X assumes the value x , add the probabilities of all the simple events for which X is equal to x Example 1 Find the probability distribution of the random variable X describing the number of heads that turn-up when a coin is flipped twice. Solution The possible values are 0, 1, and 2. P( X  0)  P(TT )  1 / 4

P( X  1)  P(TH )  P( HT )  1 / 2 P( X  2)  P( HH )  1 / 4

Requirements of discrete probability distribution If a random variable can take values x i then the following must be true:
1. 0  p(xi )  1 for all xi 2.  p ( xi )  1
all xi

The probability distribution can be used to calculate probabilities of different events. Probabilities as relative frequencies In practice, often probabilities are estimated from relative frequencies

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Example 2 The numbers of cars a dealer is selling daily were recorded in the last 100 days. This data was summarized as follows: Daily sales Frequency 0 5 1 15 2 35 3 25 4 20 100 1. Estimate the probability distribution. 2. State the probability of selling more than 2 cars a day. Solution 1. The estimated probability distribution table
x P( X  x)

0 .05

1 .15

2 .35

3 .25

4 .20

2. P(more than 2 cars a day)  P( X  2)  P( X  3)  P( X  4)  .25  .20  .45 Joint Distribution Consider two discrete random variables:  X that takes values x1 , x 2 ,..., x n  Y that takes values y1 , y 2 ,..., y m If need to see the relation ship between these two random variables the distributions of X and Y separately are not going to provide the story. For this we need the joint distribution table.

X
Y
y1 y2 … yj

x1
p11 p 21 … p j1

x2
p12 p 22 … p j2

… …

xi
p1i

… …

xn
p1n

P(Y  y)
p1 p 2 … p j

… p 2i … … … p ji … … … p mi … p i

… p2n … … … p jn … … … p mn … pn

… ym P( X  x )

… p m1
p1

… pm2
p2

… p m 1

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In this table:  (joint probability) p ji  P (Y  y j and X  xi )   (marginal probability of X ) pi  p1i  p 2i  ...  p mi  P( X  xi ) (marginal probability of Y ) p j   p j1  p j 2  ...  p jn  P(Y  y j )

Example 2a A fair coin is flipped three times. Let X be a number of heads. Y is equal to 1, if the first flip is head, and it is 0, if it is tail. Give the joint distribution of X and Y. Solution

X
Y 0 1 P( X  x )

0

1

2

3

P(Y  y)

1/8 2/8 1/8 0 1/2 0 1/8 2/8 1/8 1/2 1/8 3/8 3/8 1/8 1

The expected value Given a discrete random variable X with values x i that occur with probabilities p ( xi ) the expected value of X is

E( X ) 

x
all xi

i

 p ( xi )

The expected value of a random variable X is the weighted average of the possible values it can assume, where the weights are the corresponding probabilities of each xi. Laws of Expected Value  E (c )  c  E (cX )  cE( X )  E ( X  Y )  E ( X )  E (Y )  E ( XY )  E ( X ) E (Y ) if X and Y are independent random variables

Note: X and Y are said to be independent if for any possible values x i and y j of X and Y , respectively, we have P( X  xi , Y  y j )  P( X  xi ) P(Y  y j ) , i.e., in terms of the joint distribution p ji  p j   pi .

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Variance Let X be a discrete random variable with possible values x i that occur with probabilities p ( xi ) , and let E (X )   . The variance of X is defined to be

Var ( X )(   2 )  E ( X   ) 2  

(x
all xi

i

  ) 2 p ( xi )

The variance is the weighted average of the squared deviations of the values of X from their mean  , where the weights are the corresponding probabilities of each x i . Properties of the variance:      Var(const)=0 Var(aX)= a 2 Var(X) Var(X+Y)=Var(X)+Var(Y)+2COV(X,Y), where COV(X,Y)=E{(X-EX)(Y-EY)} If X and Y are independent, then COV(X,Y)=0 and Var(X+Y)=Var(X)+Var(Y) If Var(X)=0, then X=const

Standard deviation The standard deviation of a random variable X , denoted  , is the positive square root of the variance of X .

Example 3 The total number of cars to be sold next week is described by the following probability distribution x 0 1 2 3 4 -------------------------------------------------.35 .25 .20 p(x) .05 .15 Determine the expected value and standard deviation of random variable X , the number of cars sold. Solution E ( X )  0  .05  1  .15  2  .35  3  .25  4  .20  2.4

Var( X )  (0  2.4) 2  .05  (1  2.4) 2  .15  (2  2.4) 2  .35  (3  2.4) 2  .25  (4  2.4) 2  .2  1.24

  Var( X )  1.24  1.114

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An expected value of f (X ) is

E[ f ( X )] 

 f ( x )  p( x )
i i all xi

Bernoulli trial The Bernoulli trial can result in only one out of two outcomes. Typical cases where the Bernoulli trial applies:  A coin flipped results in heads or tails  An election candidate wins or loses  An employee is male or female  A car uses 87 octane gasoline, or another gasoline Binomial experiment 1. 2. 3. 4. There are n Bernoulli trials (n is finite and fixed). Each trial can result in a success or a failure. The probability p of success is the same for all the trials. All the trials of the experiment are independent.

Binomial Random Variable The binomial random variable counts the number of successes in n trials of the binomial experiment. By definition, this is a discrete random variable.

Calculating the Binomial Probability In general, the binomial probability is calculated by:

P( X  x)  p( x)  Cxn p x (1  p) n x
where
n! x!(n  x)! is the number of different ways of choosing x objects from a total of n objects. Here n!  1  2  3  ...  n , by convention 0!  1. C xn 

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Example 4 Suppose that we have a group of 4 people, say A, B, C, and D. How many different pairs can we select from this group? Solution The answer is “4 choose 2”: 4! 1 2  3 4 4 C2   6 2!2! 1  2  1  2 Indeed, we have six pairs: (AB), (AC), (AD), (BC), (BD), and (CD). Mean and variance of binomial random variable
E ( X )  np Var( X )  np(1  p)

Poisson Distribution The Poisson experiment typically fits cases of rare events that occur over a fixed amount of time or within a specified region Typical cases: -- The number of errors a typist makes per page -- The number of customers entering a service station per hour -- The number of telephone calls received by a switchboard per hour. Poisson Experiment Properties of the Poisson experiment:  The number of successes (events) that occur in a certain time interval is independent of the number of successes that occur in another non-overlapping time interval.  The average number of a success in a certain time interval is -- the same for all time intervals of the same size -- proportional to the length of the interval  The probability that two or more successes will occur in an interval approaches zero as the interval becomes smaller.

The Poisson Random Variable The Poisson variable indicates the number of successes that occur during a given time interval or in a specific region in a Poisson experiment Probability Distribution of the Poisson Random Variable
P( X  x )  p( x )  e  x x! x  0, 1, 2...

E ( X )  Var ( X )  

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Poisson Approximation of the Binomial    When n is very large, binomial probability table may not be available. If p is very small ( p < .05), we can approximate the binomial probabilities using the Poisson distribution. More specifically, we have the following approximation:

P( X Binomial ( p ,n )  x)  P( X Poisson ( np )  x) where the binomial distribution with parameters n and p , and Poisson with np .
The Geometric Random Variable Let X be the number of the trial on which the first success occurs in a binomial experiment.
P( X  x)  (1  p ) x 1 p , E ( X )  1 / p, Var ( X )  (1  p) / p 2

The Hypergeometric Random Variable Suppose that a population contains a finite number N of elements that possess one of two characteristics. Thus, r of the elements might be red and b  N  r , black. A sample of n elements is randomly selected from the population, and the random variable of interest is X , the number of red elements in the sample. This random variable has what is known as the hypergeometric probability distribution.

P( X  x) 

r N r Cx CN x r N r N n , E ( X )  nr / N , Var( X )  n N N N N 1 Cn

Exercises 1. Ten thousand Instant Money lottery tickets were sold. One ticket has a face value of $1000, 5 tickets have a face value of $500 each, 20 tickets are worth $100 each, 500 are worth $1 each, and the rest are losers. Let X = face value of a ticket that you buy. Find the probability distribution for X . 2. An altered die has one dot on one face, two dots on three faces, and three dots on two faces. The die is to be tossed once. Let X be the number of dots on the upturned face. Find the mean and variance of X . 3. A card is to be selected from an ordinary deck of 52 cards. Suppose that a casino will pay $10 if you select an ace. If you fail to select an ace, you are required to pay the casino $1. (a) If you play this game once, how much money does the

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casino expect to win? (b) If you play the game 26 times, how much money does the casino expect to win? 4. A bus company is interested in two potential contracts, one for an express and the other for local stops. The probabilities that the bids will be accepted are .70 and .50, with costs of $500 and $750, respectively. The estimated total incomes are $6000 and $10,000, respectively. If the company is allowed only one bid, which bid should it enter? 5. In the game of craps, a player rolls two dice. If the first roll results in a sum of 7 or 11, the player wins. If the first roll results in a 2, 3, or 12, the player loses. If the sum on the first roll is 4, 5, 6, 8, 9, or 10, the player keeps rolling until he throws a 7 or the original value. If the outcome is a 7, the player loses. If it is the original value, the player wins. The probability that a player will win is .493. Suppose that a player pays $5 to a casino if he loses and is paid $4 for a win. What is the expected loss for the player if he plays (a) one game? (b) ten games? 6. A high school class decides to raise some money by conducting a raffle. The students plan to sell 2000 tickets at $1 apiece. They will give one prize of $100, two prizes of $50, and three prizes of $25. If you plan to purchase one ticket, what are your expected net winnings? (Hint: The probability of getting the $100 ticket is 1/2000, of getting a $50 ticket is 2/2000, and of getting a $25 ticket is 3/2000.) 7. Forty percent of the students at a large university are in favor of a ban on drinking in the dormitories. Suppose 15 students are to be randomly selected. Find the probability that (a) Seven favor the ban. (b) Fewer than 4 favor the ban. (c) More than 2 favor the ban. 8. Sixty percent of the voters in a large town are opposed to a proposed development. If 20 voters are selected at random, find the probability that (a) Ten are opposed to the proposed development. (b) More than 13 are opposed to the proposed development. (c) Fewer than 10 are opposed to the proposed development. 9. Of a population of consumers, 60% are reputed to prefer a particular brand, A, of toothpaste. If a group of randomly selected consumers is interviewed, what is the probability that exactly five people have to be interviewed to encounter the first consumer who prefers brand A? At least five people?

10. In responding to a survey question on a sensitive topic (such as "Have you ever tried marijuana?"), many people prefer not to respond in the affirmative. Suppose that 80% the population have not tried marijuana and all of those individuals will truthfully answer to your question. The remaining 20% of the population have tried marijuana and 70% of those individuals will lie. Derive the probability distribution of the number of people you would need to question in order to obtain a single affirmative response.

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11. Customers arrive at a checkout counter in a department store according to a Poisson distribution at an average of seven per hour. During a given hour, what are the probabilities that a. b. c. d. No more than three customers arrive? At least two customers arrive? Exactly five customers arrive? If it takes approximately ten minutes to serve each customer, find the mean and variance of the total service time for customers arriving during a 1-hour period. Is it likely that the total service time will exceed 2.5 hours? What is the probability that exactly two customers arrive in the two-hour period of time between (a) 2:00 P.M. and 4:00 P.M. (one continuous two-hour period)? (b) 1:00 P.M. and 2:00 P.M. or between 3:00 P.M. and 4:00 P.M. (two separate one-hour periods that total two hours)?

e. f.

12. The number of typing errors made by a typist hag a Poisson distribution with an average of four errors per page. If more than four errors appear on a given page, the typist must retype the whole page. What is the probability that a randomly selected page does not need to be retyped? 13. A shipment of 20 cameras includes 3 that are defective. What is the minimum number of cameras that must be selected I fwe require that P(at least 1 defective) > .8? 14. Seed are often treated with fungicides to protect them in poor draining, wet environments. A small-scale trial, involving live treated and live untreated seeds, was conducted prior to a large-scale experiment to explore how much fungicide to apply. The seeds were planted in wet soil, and the number of emerging plants was counted. If the solution was not effective and four plants actually sprouted, what is the probability that a. All four plants emerged from treated seeds? b. Three or fewer emerged from treated seeds? c. At least one emerged from untreated seeds?


				
Lingjuan Ma Lingjuan Ma
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