# UCSB ECON 240a - Random Variables (10 pages)

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## Random Variables

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## Random Variables

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Pages:
10
School:
University of California, Santa Barbara
Course:
Econ 240a - Intro Econometrics
##### Intro Econometrics Documents

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Oct 4 2005 LEC 4 ECON 240A 1 Random Variables L Phillips I Introduction A random variable is a variable that takes on values in its range with some associated probability An example may be the number of heads in one flip of a fair coin which can take the value zero with probability or the value one with probability So a random variable is associated with a probability distribution In this example the random variable the number of heads takes on discrete values Random variables can also take on continuous values for example along the number line We will use repeated trials of a random experiment such as flipping a coin n times to study the binomial distribution Given the distribution of a random variable we will examine notions of central tendency such as the expected value of a random variable as well as measures of dispersion such as the variance of a random variable II Repeated Bernoulli Trials In the single flip of a coin heads may be the outcome with probability p or tails with probability 1 p If we consider the random variable k to be the number of heads it can take the value zero with probability 1 p or the value one with probability p The central tendency or the expected number of heads is E k i ki P ki 0 1 p 1 p p so each value of k is multiplied by its associated probability of occurrence and this weighted sum is the expected value of k If the coin is fair then the expected number of heads or mean or average value is The dispersion around the mean is the variance VAR k VAR k E k Ek 2 i ki EkI 2 P ki Oct 4 2005 LEC 4 ECON 240A 2 Random Variables L Phillips In our example the variance in the number of heads is VAR k 0 p 2 1 p 1 p 2 p p2 1 p 1 p 2 p p 1 p p 1 p p 1 p Let us expand the number of trials to two a sequence of two independent or random experiments consisting of the flip of a coin twice with outcomes given by the tree diagram in Figure 1 p p H H 1 p p 1 p T H T 1 p T Figure 1 Tree Diagram for Two Coin Flips So it is possible to get zero heads with

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