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3 The random variable X has probability density function f(;c) (which you may assume is differen-
tiable) and cumulative distribution function F(:r) where —00 &lt; :5 &lt; 00. The random variable
Y is defined by Y = X. You may assume throughout this question that X and Y have unique
modes. (i) Find the median value gm of Y in terms of the median value mm of X. (ii) Show that the probability density function of Y is f(ln y)/y, and deduce that the mode A
of Y satisfies f’(ln A) = f(ln A). (iii) Suppose now that X m N(p,a2). so that {(3): 1 e—(x—nP/(za2)_
or 2% Explain why or 211
and hence show that E(Y) = e“+%”2- 1 f” e—(z—u—a212/(2a2) d3 = 1 (iv) Show that. when X N N01, 02).
A &lt; ym &lt; E(Y).

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