ECEN4503     Exam #1         20 February 2009 

1) A random variable X has PDF f_X(x) = 0.1x; 0 < x < a.

1a) [10] Compute the value of a. [Answer: a = 4.472]

1b) [5] Compute E[X]. [Answer: 2.981]

1c) [10] Compute σ_X². [1.113]

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2) A random voltage X has PDF f_X(x) = e^-x; x > 0. This voltage is input into a square law device yielding an output Y=X².

[25] Find f_Y(y) . [f_Y(y) = e ^{-y^(0.5)}/(2y^0.5); y > 0]

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3) Two six sided fair dice are rolled, one red and one white. A random variable X = |red die -white die|. For example, if the red die is 3 and the white die is 4, x = 1.

3a) [15] Sketch f_X(x). [You should sketch f_X(x) = [6δ(x) + 10δ(x-1) + 8δ(x-2) + 6δ(x-3) + 4δ(x-4) + 2δ(x-5)]/36.]

3b) [10] Find E[X]. [1.944]

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4) You win a lottery jackpot if you correctly predict the numbers of six balls randomly drawn without replacement from an urn containing balls numbered 1, 2, 3, ... , 50. The order that the balls are drawn is not important. What matters is that the numbers you pick match the numbers drawn.

[25] Compute P(you win the jackpot). [Note: For comparison purposes, the probability that you get struck by lightning sometime during your lifetime is about 1 in 20,000.] [62.92*10^-9, i.e. 1 in 15,890,000. You're 794.5 times more likely to be struck by lightning.]

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