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Introduction. . Given a probability density function f(ur) of a continuous random variable X, we define its expected value E(X), its variance var( X) and its standard deviation (X) by r- o(X) = Vvar(X). E(X) = L =f(r) de : var(x) = [(- E(X)?f(x) dx ; In particular, these definitions are defined by improper integrals. . For a random variable X with p.d.f. f(x), we define its distribution function F(x) by F(t) = P(X < x) = (se) dt. Using the Fundamental Theorem of Calculus (FT.C.), we have that F"(x) = f(x). This provides us a way to recover the probability density function from its distribution function. For example, if we define a new random variable = ax + b where a > 0, then its distribution function is (2 - Lada "IC":(+9) L ( á e + V2πα V Fly)P(Y Sx) = P(ax + v) = P(XS f(x)dx and hence the p.d.f of Y is given by d FO) dy 1 Question 1. Suppose that a random variable X has probability density function/x() where Ho are constants and > 0 1 (a) It is known that c-dir=. By a suitable substitution, deduce that V20 (b) Compute the expected value and standard deviation of X (c) Define Y = ax + b where a, b are constants and a > 0. Find the probability density function () of Y. What is the expected value and standard deviation of Y? (d) Find constants c and d such that the random variable Z = cx + d has expected value 0 and standard deviation 1. (In this case, Z is called to have the standard normal distribution.) e