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Statistical AnalysisRandom ProcessesProbability Theory

This document from the Fall 2001 course B6014: Managerial Statistics, taught by Professor Paul Glasserman, introduces the concept of random variables and their distributions. It explains how to determine the possible values and probabilities of discrete and continuous random variables, and discusses the expected value and variance. The document also covers the concept of linear transformations and the joint distribution of multiple random variables, as well as covariance and correlation.

What you will learn

- What is a random variable and how is it denoted?
- What is the expected value of a random variable and how is it calculated for a discrete distribution?
- How do you determine the possible values and probabilities of a discrete random variable?

Typology: Slides

2021/2022

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Download Understanding Probabilities & Statistical Analysis: Random Variables, Distributions & Expe and more Slides Statistics in PDF only on Docsity! Random Variables, Distributions, and Expected Value Fall 2001 Professor Paul Glasserman B6014: Managerial Statistics 403 Uris Hall The Idea of a Random Variable 1. A random variable is a variable that takes specific values with specific probabilities. It can be thought of as a variable whose value depends on the outcome of an uncertain event. 2. We usually denote random variables by capital letters near the end of the alphabet; e.g., X,Y,Z. 3. Example: Let X be the outcome of the roll of a die. Then X is a random variable. Its possible values are 1, 2, 3, 4, 5, and 6; each of these possible values has probability 1/6. 4. The word “random” in the term “random variable” does not necessarily imply that the outcome is completely random in the sense that all values are equally likely. Some values may be more likely than others; “random” simply means that the value is uncertain. 5. When you think of a random variable, immediately ask yourself • What are the possible values? • What are their probabilities? 6. Example: Let Y be the sum of two dice rolls. • Possible values: {2, 3, 4, . . . , 12}. • Their probabilities: 2 has probability 1/36, 3 has probability 2/36, 4 has probability 3/36, etc. (The important point here is not the probabilities themselves, but rather the fact that such a probability can be assigned to each possible value.) 7. The probabilities assigned to the possible values of a random variable are its distribution. A distribution completely describes a random variable. 1 8. A random variable is called discrete if it has countably many possible values; otherwise, it is called continuous. For example, if the possible values are any of these: • {1, 2, 3, . . . , } • {. . . ,−2,−1, 0, 1, 2, . . .} • {0, 2, 4, 6, . . .} • {0, 0.5, 1.0, 1.5, 2.0, . . .} • any finite set then the random variable is discrete. If the possible values are any of these: • all numbers between 0 and ∞ • all numbers between −∞ and ∞ • all numbers between 0 and 1 then the random variable is continuous. Sometimes, we approximate a discrete random variable with a continuous one if the possible values are very close together; e.g., stock prices are often treated as continuous random variables. 9. The following quantities would typically be modeled as discrete random variables: • The number of defects in a batch of 20 items. • The number of people preferring one brand over another in a market research study. • The credit rating of a debt issue at some date in the future. The following would typically be modeled as continuous random variables: • The yield on a 10-year Treasury bond three years from today. • The proportion of defects in a batch of 10,000 items. • The time between breakdowns of a machine. Discrete Distributions 1. The rule that assigns specific probabilities to specific values for a discrete random variable is called its probability mass function or pmf. If X is a discrete random variable then we denote its pmf by PX . For any value x, P (X = x) is the probability of the event that X = x; i.e., P (X = x) = probability that the value of X is x. 2. Example: If X is the outcome of the roll of a die, then P (X = 1) = P (X = 2) = · · · = P (X = 6) = 1/6, and P (X = x) = 0 for all other values of x. 2 6. The variance of a random variable X is denoted by either V ar[X] or σ2 X . (σ is the Greek letter sigma.) The variance is defined by σ2 X = E[(X − µX)2]; this is the expected value of the squared difference between X and its mean. For a discrete distribution, we can write the variance as σ2 X = ∑ x (x − µX)2P (X = x). 7. An alternative expression for the variance (valid for both discrete and continuous random variables) is σ2 X = E[(X2)]− (µX)2. This is the difference between the expected value of X2 and the square of the mean of X. 8. The standard deviation of a random variable is the square-root of its variance and is denoted by σX . Generally speaking, the greater the standard deviation, the more spread-out the possible values of the random variable. 9. In fact, there is a Chebyshev rule for random variables: if m > 1, then the probability that X falls within m standard deviations of its mean is at least 1− (1/m2); that is, P (µx − mσX ≤ X ≤ µX + mσX) ≥ 1− (1/m2). 10. Find the variance and standard deviation for the roll of one die. Solution: We use the formula V ar[X] = E[X2] − (E[X])2. We found previously that E[X] = 3.5, so now we need to find E[X2]. This is given by E[X2] = 6∑ x=1 x2PX(x) = 12( 1 6 ) + 22( 1 6 ) + · · ·+ 62( 1 6 ) = 15.167. Thus, σ2 X = V ar[X] = E[X2]− (E[X])2 = 15.167 − (3.5)2 = 2.917 and σ = √ 2.917 = 1.708. Linear Transformations of Random Variables 1. If X is a random variable and if a and b are any constants, then a + bX is a linear transformation of X. It scales X by b and shifts it by a. A linear transformation of X is another random variable; we often denote it by Z. 2. Example: Suppose you have investments in Japan. The value of your investment (in yen) one month from today is a random variable X. Suppose you can convert yen to dollars at the rate of b dollars per yen after paying a commission of a dollars. What is the value of your invenstment, in dollars, one month from today? Answer: −a + bX. 5 3. Example: Your salary is a dollars per year. You earn a bonus of b dollars for every dollar of sales you bring in. If X is what you sell, how much do you make? 4. Example: It takes you exactly 16 minutes to walk to the train station. The train ride takes X hours, where X is a random variable. How long is your trip, in minutes? 5. If Z = a + bX, then E[Z] = E[a + bX] = a + bE[X] = a + bµX and σ2 Z = V ar[a + bX] = b2σ2 X . 6. Thus, the expected value of a linear transformation of X is just the linear transformation of the expected value of X. Previously, we said that E[g(X)] and g(E[X]) are generally different. The only case in which they are the same is when g is a linear transformation: g(x) = a + bx. 7. Notice that the variance of a + bX does not depend on a. This is appropriate: the variance is a measure of spread; adding a does not change the spread, it merely shifts the distribution to the left or to the right. Jointly Distributed Random Variables 1. So far, we have only considered individual random variables. Now we turn to properties of several random variables considered at the same time. The outcomes of these different random variables may be related. 2. Examples (a) Think of the price of each stock in the New York exchange as a random variable; the movements of these variables are related. (b) You may be interested in the probability that a randomly selected shopper buys prepared frozen meals. In designing a promotional campaign you might be even more interested in the probability that that same shopper also buys instant coffee and reads a certain magazine. (c) The number of defects produced by a machine in an hour is a random variable. The number of hours the machine operator has gone without a break is another random variable. You might well be interested in probabilities involving these two random variables together. 3. The probabilities associated with multiple random variables are determined by their joint distribution. As with individual random variables, we distinguish discrete and contin- uous cases. 6 4. In the discrete case, the distribution is determined by a joint probability mass func- tion. For example, if X and Y are random variables, there joint pmf is PX,Y (x, y) = P (X = x, Y = y) = probability that X = x and Y = y. For several random variables X1, . . . ,Xn, we denote the joint pmf by PX1,...,Xn . 5. It is often convenient to represent a joint pmf through a table. For example, consider a department with a high rate of turnover among employees. Suppose all employees are found to leave within 2-4 years and that all employees hired into this department have 1-3 years of previous work experience. The following table summarizes the joint probabilities of work experience (columns) and years stayed (rows): 1 2 3 2 .03 .05 .22 3 .05 .06 .15 4 .14 .15 .15 Thus, the proportion of employees that had 1 year prior experience and stayed for 2 years is 0.03. If we let Y = years stayed and X = years experience, we can express this as PX,Y (1, 2) = P (X = 1, Y = 2) = 0.03. The table above determines all values of PX,Y (x, y). 6. What proportion of employees stay 4 years? What proportion are hired with just 1 year of experience? These are questions about marginal probabilities; i.e., probabilities involving just one of the random variables. A marginal probability for one random variable is found by adding up over all values of the other random variable; e.g., P (X = x) = ∑ y P (X = x, Y = y), where the sum ranges over all possible y values. In the table, the marginal probabilities correspond to the column-sums and row-sums. So, the answers to the two questions just posed are 0.44 and 0.22 (the last row-sum and the first column-sum). 7. From a joint distribution we also obtain conditional distributions. The conditional distribution of X given Y = y is PX|Y (x|y) = P (X = x|Y = y) = P (X = x, Y = y) P (Y = y) . To find a conditional distribution from a table, divide the corresponding row or column by the row-sum or column-sum. 7