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Chapter 4 Discrete Random Variables 4. 1 4. 2 4. 3 4. 4 4. 5 4. 6 Discrete and Continuous Probability Distribution Expectation and Variance Binominal Poisson Hypergeometric Homework: 3, 5, 7, 9, 11, 13, 15, 21, 23, 27, 31, 43, 52, 53, 61, 63, 68, 69, 77 1
Last chapter we discussed several useful concepts of dealing with probability problems. However, it is very difficult to write down sample spaces of some random experiments. In these cases the concept random variable is very useful. A random variable is a variable that assumes values associated with the random outcomes of a random experiment, where one and only one numerical values is assigned to each sample point. It is random because we can not predict the outcome of a random experiment. It is a variable because there are more than one possible sample points in a random experiment. 2
Section 4. 1: Discrete and Continuous Random Variable Some random variable can assume values on countable many numbers (such as integers) and some random variable can assume values on one or more intervals. For example, the distance between you home and UCF is between 0 and 100 miles that is an interval, i. e. the distance between your home and UCF is a continuous random variable. But the number of head in coin tossing experiment is a countable number, i. e. the number of heads in a coin tossing experiment is a discrete random variable. 4
Sec 4. 2: Probability Distributions for Discrete Random Variables This chapter will focus on the discussion of discrete random variable. A complete description of a discrete random variable requires that we specify all the possible values the random variable can assume and the probability associated with each value. Usually, we can use the following four steps to complete a probability table. Step 1: Find out the variable of interest. Step 2: List all the sample points in the sample space. Step 3: List all the possible values of this random variable. Step 4: Assign the probabilities to all the possible values. 6
The probability distribution of a discrete random variable is a graph, a table, or a formula that specifies the probability associated with each possible value the random variable can assume. The probability distribution should not include values that have zero probabilities. The rules to assign probability discussed in Section 3. 2 should be followed as well. Thus, the probability of any value of a random variable is between 0 and 1 and the sum of the probabilities of all possible values of a random variable is equal to one. 8
Sec 4. 3: Expectation and Variance of a Discrete Random Variable We discussed how to obtain the sample mean, the sample variance, and the sample standard deviation in chapter 2. Now, we introduce the formulas of getting the population mean, the population variance, and the population standard deviation of a discrete random variable. Suppose X is a discrete random variable with probability distribution p(x). The expectation of X is the population mean of X. Let m, , and s be the population mean, the population variance, and the population standard deviation of X, respectively. Then m = E(x) = S xp(x), 2 2 2 s = E[(x-m) ] = S (x-m) p(x), and 11
(a) Find the expectation of this random variable.
(b) Find the standard deviation of this random variable.
(c) What is the probability that x falls within the interval (m-2 s, m+2 s)? (d) Does the result satisfy the Chebyshev’s Rule? (e) Does the result satisfy the Empirical Rule? Explain.
(a) Write down the probability table.
(b) Find the mean.
(c) Find the standard deviation of the game.
(d) Do you believe the lottery officials claim ``the more you play the more you win’’?
(b) Find the mean and standard deviation.
(c) Find P(x >= 9). (d) Can you apply the Empirical rule to find the probability of X falls into the interval (m-2 s, m+2 s)?
Sec 4. 4: The Binomial Random Variable The responses of many experiments have only two alternatives such as "Yes or No”, "True or False", "Male or Female”, and "Failure or Success". These types of experiments have some characteristic in common. First, they consist of n identical and independent trials. Second, there are only two possible outcomes, denoted by S and F on each trail. Third, the possibility of each outcome remains unchanged from trial to trial, that is, the probability of S is p and probability of F is q=(1 -p) in each trial. 24
Fourth, we are interested in the random variable x represented the number of S happened in n trails (n is a fixed number). Therefore, it is worth to develop a special probability model to deal with this kind of random variables. Any random variable that has these four characteristics is called binomial random variable and can be dealt with by using this special probability model. 25
Suppose that X is a binomial random variable. The probability of success on any single trial is p and there are n trials in this random experiment. The probability density function of X is where p = the probability of success on any single trial n = total number of trials q=1 -p x = number of successes in n trials. 28
Let m and s be the mean and standard deviation of the binomial random variable X. In stead of using the expectation summation rules to calculate m and s, we can find m and s easily using the formulas m = np, s**2 = npq = np(1 -p), and 29
(b) Write down the probability variable.
(c) Find the mean and standard deviation of this random variable.
(b) If the firm will be able to stay in business only if two or more holes produce oil, what is the probability that it can survive.
Note: (1) We can not use the Binomial probabilities Table to obtain this probability because p = 0. 12 is not in the Table. (2) We can obtain this probability much easier with the concept of complement event: P(X 2) = 1 - P(X 1) = 1 - P(X=0) - P(X=1) = 1 - 0. 4644044086 - 0. 37996698 @ 0. 156 36
Section 4. 5: The Poisson Random Variable The random variables produced by many random experiments can be well described by using Poisson probability model. Typical examples are as follows: (1) the number of customers served per hour in a given restaurant, (2) the number of alcohol related traffic accidents per month at a busy intersection, (3) the number of diseased trees per acre in a certain national park, (4) the number of telephone calls received per minute during your lunch hour. 37
Poisson random variable has the following common characteristics. (1). The experiment consists of counting the number of times that a certain event occurs during a given unit of time or in a given area or volume. (2). The probability of an event occurs in a given unit of time is same for all time units. (3). The number of events that occur in one unit of time, area, or volume is independent of the number that occur in other units. 38
The probability density function of a Poisson random variable is Both the mean and the variance of a Poisson random variable equals to l, i. e. m = l and s 2 = l. 39
(b) P(x 2) when m = 2.
(c) P(x >3) where s 2 = 3.
Note: For Poisson random variable, we can find P(x a) from the table directly if a is an integer. We need to apply the concept of complement event to find the probability of P(x > a) or P(x a). We need to know that P(x > a) = 1 - P(x a) and P(x a ) = 1 - P(x a-1). 43
Section 4. 6 The Hypergeometric Random Variable Hypergeometric random variable is another popular discrete random variable. Suppose there a total of N balls: r red balls and (N-r) white balls, in a bag. And n balls are randomly selected from this bag without replacement. Let X denote the number of red balls in the n balls selected. Then the distribution of X is called a Hypergeometric distribution, with parameters N, r and n. 47
The probability density function of this hypogeometric distribution is given by: 48*
(b) N=9, n=5, r=3, x=3.
Collection of Definitions: Random Variable A random variable is a rule that assigns one and only one numerical value to each sample point in a random experiment. Discrete Random Variable A discrete random variable is a random variable that can assume only countable number of values. Continuous Random Variable A continuous random variable is a random variable that can assume values in one or more intervals. 51
Probability Distribution The probability distribution of a discrete random variable is a way, such as a graph, a table, or a formula, that specifies the probability associated with each possible value the random can assume. Expectation of a Discrete Random Variable The expectation of a discrete random variable is the population mean of this random variable. We can use the following formula to compute the expectation of a discrete random variable m = E(x) = S xp(x). 52
Variance of Discrete Random Variable The variance of a discrete random variable is the population variance of the random variable, given by formula 2 2 2 s = E[(x-m) ] = S (x-m) p(x). Standard Deviation of Discrete Random Variable The standard deviation of a discrete random variable is equal to the square root of the variance of this random variable, i. e. 53
Binomial Distribution The probability density function of a binomial random variable is Here p = the probability of success on any single trial ; n = total number of trials; x = number of successes in n trials; q = 1 - p; The mean of a binomial random variable is np, i. e. m = np; The variance of a binomial random variable is npq, i. e. s**2 = npq = np(1 -p). 54
Poisson Random Variable The probability density function of a Poisson random variable is Both the mean and the variance of a Poisson random variable equals to l, i. e. m = l and s**2 = l. 55
Hypergeometric Random Variable The probability density function of a Hypergeometric random variable is where N = total number of balls in the bag; r = the number of red balls in the bag; n = the number of balls drawn without replacement; x = the number of red balls in the n balls selected. 56
The mean of a Hypergeometric random variable is and the variance of a Hypergeometric random variable is 57