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Lecture 10

Lecture 10.docx

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McMaster University
Jeff Racine

Lecture 10 Mean of a discrete random variable: the mean of a discrete random variable (i.e., the average or “expected value”) is denoted by E(X), µ ,xor simply µ. The expected value is the weighted average of all possible values of X, where the weights are the probabilities associated with the particular X values, i.e., n E(X)= ∑ xi× p(X=x ) i i=1 , Where the sum is taken over all possible value of X Expected value of a function of a random variable: Let X be a discrete random variable, and let Y be a function of X such that Y = g(X). Then the expected value of Y (the expected value of g(X)) is n E(Y)=E g([)= ] g(x )× p(X=x ) i=1 i i Example Consider a game of chance (i.e., “random experiment”) where it is possible to lose $1.00, break even, win $3.00, or win $5.00 each time one plays The probability distribution for each outcome is as follows: Outcome [x] i -$1.00 $0.00 $3.00 $5.00 Probability 0.3 0.4 0.2 0.1 [p(xi] The expected outcome (average, mean) for this game can be computed as follows: Variance of a discrete random variable: 2 n 2 σ x= ∑ (xi−μ ) (=x i i=1 Note that the variance denotes the average squared deviation from the mean, and is 2 2 referred to as E[(X−μ) ] . Using the rule E[g(X)] =E [(X)=]∧g(x) = (x−μ) , we obtain the formula for The Binomial Probability Distribution The ‘binomial’ distribution is one of the most useful discrete distributions we shall use A binomial probability distribution shows the probabilities associated with possible values of a discrete random variable that is generated by a type of experiment called a Bernoulli process A Bernoulli process consists of a sequence of n identical trials of a random experiment such that each trial Produces one of two possible complementary outcomes that are conventionally called success and failure Stands independent of any other trial so that the probability of success or failure is constant from trial to trial The number of successes achieved in the process is the binomial random variable Let X denote the number of successes obtained in n independent trials in which the probability of a success on any trial is π. The
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