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Chapter 9

ECO220Y1 Chapter 9 Notes

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Jennifer Murdock

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ECO220Y1 Textbook Notes Chapter 9: Random Variables and Probability Distributions 9.1 Expected Value of a Random Variable  Random variable: assumes any of several different values as a result of some random event. o Denoted by a capital letter, such as X. o The value it may take on can be denoted by the corresponding lowercase letter, for example, x.  Discrete random variable: a random variable that can take one of a finite number of distinct outcomes. o I.e. either 10, 20, 30, 40, or 50 etc.  Continuous random variable: a random variable that can take any numeric value within a range of values. The range may be infinite or bounded at either or both ends. o I.e. 10 < x < 20  Probability distribution: a function that associates a probability P with each value of a discrete random variable X, denoted P(X = x), or with any interval values of a continuous random variable. o Also called a probability model.  Expected value (mean): the expected value of a random variable is its theoretical long-run average value, the centre of its model. Denoted μ or E(X), it’s found (if the random variable is discrete) by summing the products of variable values and probabilities: ( ) ∑ ( ) o It isn’t what actually happens to any particular policyholder; it is the average amount. 9.2 Standard Deviation and Variance of a Random Variable  Standard deviation of a (discrete) random variable: describes the spread in the model and is the square root of the variance. ( ) √ ( ) √ ∑( ( )) ( ) 9.3 Adding and Subtracting Random Variables  Changing a random variable by a constant ( ) ( ) ( ) ( ) ( ) ( ) Adding/subtracting a constant will shift the data but not change the variance and therefore won’t change the SD. ( ) ( ) ( ) ( ) ( ) | | ( )  Addition Rule for Expected Values of Random Variables: the expected value of the sum (or difference) of random variables is the sum (or difference) of their expected values. ( ) ( ) ( )  Fundamental principle behind insurance is that by spreading risk among many policies, a company can keep the SD quite small and predict costs more accurately.  Addition Rule for Variances of (Independent) Random Variables: the variance of the sum or difference of two independent random variables is the sum of their individual variances. ( ) ( ) ( ) ( ) √ ( ) ( )  The variance of one policyholder with an expected payout that is doubled, is 4x the original variance (2 V[X]) whereas the variance of two policyholders with the original expected payout is just 2x the original variance (V[X] + V[Y]). o Thus the SD is higher with the one policyholder who has a doubled possible payout vs. the two policyholders.  Summary: o When we subtract the variables, E – I, we add the variances, Var(E) + Var(I). o SD’s don’t add or subtract. The SD is obtained by taking the square root of the variance. o The expected value of the sum of two random variables is the sum of the expected values. o The expected value of the difference of two random variables is the difference of the expected values. o If the random variables are independent, the variance of their sum or difference is always the sum of the variances.  The sum or difference of the variances of correlated variables is: ( ) ( ) ( ) ( ) ( )  The sum or difference of the variances of variables multiplied by constants is: ( ) ( ) ( ) ( ) ( )  The sum or difference of the expected values of the variables multiplied by constants is: ( ) ( ) ( )  When r < 0, the variance (riskiness of an investment) is reduced. o “Don’t put all your eggs in one basket.” 9.4 Discrete Probability Distributions  If X is a random variable with possible outcomes 1, 2, … , n and P(X = i) = 1/n for each value of I, then we say X has a discrete Uniform Distribution.  By convention, one of the outcomes in a trial is denoted a “success” and the other a “failure”. o Often the less common outcome or the one that calls for action is called a success.  Bernoulli trials: a sequence of trials is called Bernoulli if: o There are exactly two possible outcomes (usually denoted success and failure).  I.e. webpage loads correctly (failure) or does not (success). o The probability of success is constant.  It is denoted p and the probability of failure is denoted q = 1 – p. o The trials are independent.  I.e. the fact that one page loaded correctly will not impact if the next will or not.  10% condition: we are ok to proceed in using a probability distribution to model different aspects of Bernoulli trials so long as the sample taken is smaller than 10% of the population. o Also requires the outcomes to be independent. 9.6 The Binomial Distribu
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