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Unformatted text preview: ECEN 303: Assignment 10
Problems: 1. Let X be a random variable with PDF fX (x) = and let A be the event {X ≥ 2}. (a) Find E[X ], Pr(A), fX A (x), and E[X A]. (b) Let Y = X 2 . FInd E[Y ] and Var[Y ]. 2. We start with a stick of length ℓ. We break it at a point which is chosen according to a uniform distribution and keep the piece, of length X , that contains the left end of the stick. We then repeat the same process on the piece that we were left with, and let Y be the length of the remaining piece after breaking for the second time. (a) Find the joint PDF of X and Y . (b) Find the marginal PDF of Y . (c) Use the PDF of Y to evaluate E[Y ]. (d) Evaluate E[Y ], by exploiting the relation Y = X · (Y /X ). 3. A defective coin minting machine produces coins whose probability of heads is a random variable P with PDF pep , p ∈ [0, 1], fP (p) = 0, otherwise. A coin produced by this machine is selected and tossed repeatedly, with successive tosses assumed independent. (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 . (c) Given that a ﬁrst coin toss resulted in heads, ﬁnd the conditional probability of heads on the next toss. 4. Let X be a random variable that takes the values 1, 2, and 3, with the following probabilities: 1 Pr(X = 1) = , 2 1 Pr(X = 2) = , 4 1 Pr(X = 3) = . 4 x/4, if 1 < x ≤ 3, 0, otherwise, Find the tranform (moment generating function) associated with S and use it to obtain the ﬁrst three moments, E[X ], E X 2 , E X 3 . 1 5. Find the PDF of the continuous random variable X associated with the transform (moment generating function) 1 2 2 3 M (s) = · +· . 3 2−s 3 3−s 6. Suppose that X is a standard normal random variable. (a) Calculate E X 3 and E X 4 . (b) Deﬁne a new random variable Y such that Y = a + bX + cX 2 . Find the correlation coeﬃcient ρ(X, Y ). 2 ...
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 Fall '07
 Chamberlain

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