Inference based on discrete observations Example The treasury of an

# Inference based on discrete observations example the

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Inference based on discrete observations . Example . . The treasury of an underdeveloped country produces coins whose probability of heads is a r.v. P with PDF f P ( p ) = pe p if p [ 0 , 1 ] 0 , o.w. . (a) Find the probability that a coin toss results in heads. (b) Given that a coin toss resulted in heads, find the conditional PDF of P . (Exercise: What if the coin is tossed twice and heads is observed once?) (c) Given that the first coin toss resulted in heads, find the conditional probability of heads on the next toss. 74/75
Continuous Random Variables and PDFs Cumulative Distribution Functions Normal Random Variables Multiple Continuous Random Variables Conditioning The Continuous Bayes’ Rule Inference about a cont. r.v. based on cont. observations Inference about a discrete r.v. Inference based on discrete observations . Some Additional Problems . Example (The sum of a random number of r.v.s) . . You visit a random number N of stores and in the i th store, spend a random amount of money X i . The total amount of money you spend is T = X 1 + X 2 + . . . + X N where N is a positive integer r.v. with mean E [ N ] and variance var ( N ) . The r.v.s X i are independent and identically distributed (i.i.d.), and have mean E [ X ] and variance var ( X ) . Evaluate the mean and variance of T . 75/75

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