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by: Rahul Bose

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# Econometrics: Random Variables ECO 5100

Marketplace > Wayne State University > Economcs > ECO 5100 > Econometrics Random Variables
Rahul Bose
WSU

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COURSE
Econometrics
PROF.
Arjun
TYPE
Class Notes
PAGES
2
WORDS
KARMA
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## Popular in Economcs

This 2 page Class Notes was uploaded by Rahul Bose on Wednesday June 22, 2016. The Class Notes belongs to ECO 5100 at Wayne State University taught by Arjun in Summer 2016. Since its upload, it has received 5 views. For similar materials see Econometrics in Economcs at Wayne State University.

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Date Created: 06/22/16
Chapter 2 : Random variables Consider the problem of an airline trying to decide how many reservations to accept for a flight that has 100 available seats. If fewer than 100 people want reservations, then these should all be accepted. But what if more than 100 people request reservations? A safe solution is to accept at most 100 reservations. However, becasue some people book reservations and then do not show up for the flight, there is some chance that the plane will not be full even if 100 reservations are booked. A different strategy is to book more than 100 reservations and to hope that some people do not show up, so the final number of passengers is as close to 100 as possible. This policy runs the risk of the airline having to compensate people who are necessarily bumped from an overbooked flight. Can we decide the optimal number of reservations that the airline should make? Given certain information (on airline costs and how frequently people show up for reservations), we can use basic probability to arrive at a solution. Suppose that we flip a coin 10 times and count the number of times the coin turns up heads. This is an example of an experiment. Generally, an experiment is any procedure that can be infinitely repeated and has a well- defined set of outcomes. We could carry out the coin-flipping procedure again and again, and we know that the number of heads appearing is an integer from 0-10. A random variable is one that takes on numerical values and has an outcome that is determined by an experiment. In the coin-flipping example, the number of heads appearing in 10 flips of a coin is an example of a random variable. Before we flip the coin 10 times, we do not know how many times the coin will come up heads. Once we flip the coin 10 times and count the number of heads, we obtain the outcome of the random variable for this particular trial of experiment. Another trial can produce a different outcome. In the airline reservation example, the number of people showing up for their flight is a random variable: before any particular flight, we do not know how many people will show up. We denote random variables by uppercase letters, usually W, X, Y and Z; particular outcomes of random variables are denoted by the corresponding lowercase letters, w, x, y and z. e.g. in the coin-flipping experiment, let X denote the number of heads appearing in 10 flips of a coin. Then, X is not associated with any particular value, but we know X will take on a value in the set 0,1,2,...,10. A particular outcome is, say, x = 6. We indicate large collections of random variables by using subscripts. e.g. if we record last year’s income of 20 randomly chosen households in the US, we might denote these random variables by X1, X2, ..., X20, the particular outcomes would be denoted x1, x2, ..., x20. Xu Lin (Wayne State University) Chapter 2: Random Variables and Probabilit Random variables are always defined to take on numerical values, even when they describe qualitative events. e.g. consider tossing a single coin, where the two outcomes are heads and tails. We can define a random variable as: X = 1 if the coin turns up heads, and X = 0 if the coin turns up tails. A random variable that can only take on the values zero and one is called a Bernoulli (or binary)random variable. In basic probability, it is traditional to call the event X = 1 a "success" and the event X = 0 a "failure". A discrete random variable is one that takes on only a finite or countably infinite number of values. The notion of "countably infinite" means that even though an infinite number of values can be taken on by a random variable, those values can be put in a one-to-one correspondence with positive integers. A Bernoulli random variable is the simplest example of a discrete random variable. The only thing we need to completely describe the behavior of a Bernoulli random variable is the probability that it takes on the value one. In the coin-flipping example, if the coin is "fair", then P(X = 1) = 1/2 (read as "the probability that X equals one is one-half"). And P(X = 0) = 1/2, also. In the previous airline example, for a randomly selected customer, we define a Bernoulli random variable as X = 1 if the person shows up for the reservation, and X = 0 if not. Letting P(X = 1) = θ P(X = 0) = 1 − θ e.g. θ = 0.75. The value of θ is crucial in determining the airline’s strategy for booking reservations. We will see how to estimate θ later. Notation: X ∼Bernoulli(θ) is read as "X has a Bernoulli distribution with probability of success equal to θ”. More generally, any discrete random variable is completely described by listing its possible values and the associated probabilities that it takes on each value. If X takes on the k possible values {x1, ..., xk}, then the probabilities p1, p2, ..., pk are defined by pj = P(X = xj ), j = 1, 2, ..., k where each pj is between 0 and 1 and p1 + p2 + ... + pk = 1.

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