hw4-2011 - a Verify that this is indeed a joint density...

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Theory of Probability, HW #4, all sections Assignment Date: April 22, 2011; Due Date: April 29, 2011 1. Two types of coins are produced in a factory: a fair coin and a biased one that comes up heads 55 percent of the time. We have one of these coins but do not know which one. In order to ascertain which type of coin we have, we shall perform the following statistical test: We shall toss the coin 1000 times. If the coin lands on heads 525 or more times, then we shall conclude that it is a biased coin, whereas, if it lands on heads less than 525 times, then we shall conclude that it is a fair coin. a. If the coin is actually fair; what is the probability that we shall reach a false conclusion? b. What would the probability be if the coin were biased? 2. The joint probability density function of X and Y is given by ±²³ ´µ ¶· ¸ ¹ º² » ¼· ²´ ½ ¾ ··········¿ À ² À Á³·············¿ À ´ À ½
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Unformatted text preview: a. Verify that this is indeed a joint density function. b. Compute the density function of X . c. Find P [ X > Y ] d. Find P [ Y > ½| X < ½] e. Find E[ X ] and E[ Y ]. 3. If X and Y are independent exponential random variables with parameter λ = 1, compute the joint density of a. U = X + Y , V = X / Y ; b. U = X , V = X / Y ; c. U = X + Y , V = X / ( X + Y ). 4. A point is chosen at random (according to uniform a PDF) within the semicircle of the form  à ±²³ ´µÄ·² » ¼·´ » ŠƳ ´ Ç ¿È³· for some given r > 0. a. Find the joint PDF of the coordinates X and Y of the chosen point. b. Find the marginal PDF of Y and use it to find E[ Y ]. 5. The random variables X , Y , and Z are independent and uniformly distributed between zero and one. Find the PDF of X + Y + Z ....
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