PS1 Solutions

# PS1 Solutions - Department of Economics University of...

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Department of Economics Spring 2011 University of California Prof. Woroch Economics 140: Problem Set 1 SUGGESTED ANSWERS 1. You have a standard deck of 52 playing cards consisting of the usual 13 cards of each of 4 suits (hearts, diamonds, clubs, spades). You shuffle the deck and draw a card and note its suit. You put that card back in the deck, re-shuffle the deck, and draw another card and note its suit. Let M be the random variable that equals the sum of the spades drawn. The sample space of this “experiment” includes 0, 1 and 2 spades. a) Are the two draws independent and identically distributed (i.e., i.i.d.)? Explain. Presuming that the deck is shuffled well before each draw, the second draw will be independent of the first draw. Since the first card drawn is put back into the deck before the second draw, the second draw has the same distribution as the first. If this was “sampling without replacement” then it would not be i.i.d. b) Express the probability density function (p.d.f.) and the cumulative distribution function (c.d.f.) of the random variable M. Each draw is independent and in each one of them the probability of drawing a spade is 1/4. Let S represent a spade and N represent any other suit. We have the following events possible and respective probabilities. ܵܵ , ܲ ( ܵܵ ) = 1 16 ܵܰ , ܲ ( ܵܰ ) = 3 16 ܰܵ , ܲ ( ܰܵ ) = 3 16 ܰܰ , ܲ ( ܰܰ ) = 9 16 Probability distribution: ܲ ( ܯ = 2) = ܲ ( ܵܵ ) = 1 16 ܲ ( ܯ = 1) = ܲ ( ܵܰ ) + ܲ ( ܰܵ ) = 6 16 ܲ ( ܯ = 0) = ܲ ( ܰܰ ) = 9 16 Cumulative probability: ܲ ( ܯ ≤ 0) = ܲ ( ܯ = 0) = 9 16 ܲ ( ܯ ≤ 1) = ܲ ( ܯ ≤ 0) + ܲ (= 1) = 15 16 ܲ ( ܯ ≤ 2) = ܲ ( ܯ ≤ 1) + ܲ ( ܯ = 2) = 1 c) Give the population mean and variance of M. ܧ ( ܯ ) = 0 ܲ ( ܯ = 0) + 1 ܲ ( ܯ = 1) + 2 ܲ ( ܯ = 2) = ଵ଺ + 2 ଵ଺ = ଵ଺ = . ܸ ( ܯ ) = ܧ ( ܯ ) − ܧ ( ܯ ) = 0 ܲ ( ܯ = 0) + 1 ܲ ( ܯ = 1) + 2 ܲ ( ܯ = 2) = ଵ଺ + 4 ଵ଺ = ଵ଺ . d) Using Excel, generate 50 trials of this simple experiment (i.e., draw two cards with replacement and sum the number of spades drawn). [Hint: use the =RAND() function in Excel to draw a card randomly by classifying a draw from 0 through 0.25 as a spade, 0.25 through 0.5 as a diamond, and so on.]

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In the Excel file. e)
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PS1 Solutions - Department of Economics University of...

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