mid2F03

mid2F03 - T exceeds 35(Hint use Φ t ≈ 1 2 t(4 4-t 10 0...

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McGill University, Faculty of Engineering Course ECSE-305A: Probability and Random Signals I Midterm Examination #2, Fall 2003 Date and time: Thursday, November 6, 2003, 14:35 - 15:55 Examiner: Prof. B. Champagne Instructions: This is a closed book examination: only the faculty standard calculator is allowed. Attempt all questions. 1. Consider a sequence of independent Bernouilli trials, each with probability of success p , 20 marks where 0 < p < 1. Let X be the number of experiments until the ﬁrst success occurs. (a) With the help of a tree diagram, show that the PMF of X is given by P ( X = n ) = p (1 - p ) n - 1 , n = 1 , 2 , 3 ,... (b) Let k be a non-negative integer. Find a compact expression for P ( X k ). (c) Using the identity x =1 xr x = r/ (1 - r ) 2 , where | r | < 1, ﬁnd the expected value of X . 2. Suppose that the temperature T , measured in o C at noon time in Montreal during the 20 marks month of July is a normal RV with variance 25; also assume that P ( T 30) = 1 / 2. (a) Find the mean value of T . (b) Find the probability that
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Unformatted text preview: T exceeds 35. (Hint: use Φ( t ) ≈ 1 2 + t (4 . 4-t ) 10 , 0 ≤ t ≤ 2 . 2). (c) What is the probability that the temperature exceeds 35 on at least one day during the month of July? (Important hint: there are 31 days in July). 3. Let X be a continuous RV with PDF 20 marks f ( x ) = ( 2 | x | 3 ,-1 < x < 1 , otherwise Using the method of transformation, ﬁnd g ( y ), the PDF of RV Y = 1-X 2 . 4. Let X be a Laplacian RV with parameter α > 0. The PDF of X is given by 20 marks f ( x ) = α 2 e-α | x | , x ∈ R (a) Find the characteristic function of X . (b) Using the result in part (a), ﬁnd E ( X ) and E ( X 2 ). 5. Let X and Y be jointly continuous with joint PDF 20 marks f ( x,y ) = ( 4 xy if 0 ≤ x ≤ 1 and 0 ≤ x ≤ 1 , otherwise . (a) Compute the probability that Y ≥ X . (b) Find the marginal PDF of X and sketch it. 1...
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