MATH-0042 Midterm: w12exam2sol
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2 e (x )2/2 2: a partial list of approximate values of the function. Your answer should be a number; justify it by writing an integral expression that represents this probability and showing how to nd its value. (5 points) the probability that the random variable x is positive is given by: To realize this quantity in terms of f , we rst make a standardizing substitution. Let t = x so that dt = 1 dx and the limits change as follows: x = 0 t = One way to proceed from here is to use the symmetry substitution u = t (so that du = dt and the signs of the limits ip); we nd. 2 e u2/2 du = f (0. 5) 0. 69. Alternatively, we may use the relation f ( z) = 1 f (z) (also a symmetry ; see part (b) for more details) to nd that. = 1 (1 f (0. 5)) = f (0. 5) 0. 69.
