03Summer-HE2

03Summer-HE2 - 1,4], and let Y = X 1 (a) Find E[ Y ]. (b)...

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University Second Hour Exam ECE 313 of Illinois Summer 2003 1. Check the appropriate box for each statement below. Answers need not be justified. However, in order to discourage guessing, you will be penalized for wrong answers. Which of the following statements are true for all CDFs? TRUE FALSE P{ X > b} = 1 F X (b) If a < b, then F X (a) < F X (b) lim a →∞ F X (a) = 1 Let f X (u) denote the probability density function (PDF) of a continuous random variable X . Which of the following statements are true for all PDFs? TRUE FALSE f X (u) 1 for all u, −∞ < u < P{a < X < b} = P{a < X b} lim u →−∞ f X (u) = 0 2. X denotes a Gaussian random variable with mean 2 and variance 16. Find P{ | X – 4 | > 3} and P{ X < 3 | X > 2} using the table of values of the unit Gaussian CDF Φ (•). 3. Let X denote a random variable uniformly distributed on the interval [
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Unformatted text preview: 1,4], and let Y = X 1 (a) Find E[ Y ]. (b) Find f Y (v), the probability density function of Y . 4. Consider a Poisson process with arrival rate 2 per second. Let A denote the event that there are no arrivals in the time interval (0, T] and B the event that there is exactly one arrival in the time interval (0.5T, 1.5T]. (a) What are the values of P(A) and P(B)? (b) Find P(B | A). 5. A coin is tossed once, and heads shows. Assuming that the probability p of heads is the value of a random variable X uniformly distributed in the interval (0.4,0.6), find the probability that at the next tossing heads will show. ( Hint: The asked probability is the Bayesian estimate of probability of heads after the observation)....
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