Probability and Statistics II week 4

Properties of Expectation Let X be a random variable and c a constant. Then we have: (i) E (c) = c, for c a constant (ii) E [c g (x)] = c E [g (x)], where c is a constant (iii) E [c1 g1 (x) + c2 g2 (x)] = c1 E [g1 (x)] + c2 E [g2 (x)] where c1 and c2 are constants. Proof We shall assume that the random variable X is continuous with probability density function . (i) since is a p.d.f. (ii) . (iii) By definition; If the random variable X is discrete, replace integral notations with summation notations.   2.2. VARIANCE Let X be a random variable with mean .The variance of X denoted by Var (x), or , is given by Var (X) = Hence Example 2.4 Let X be a random variable with the following probability distribution Obtain the variance of X. Solution Evaluating E (x2), we get Example 2.5 Consider tossing two dice once and considering the sum of upturned faces. Let X be the sum of upturned faces when the two dice are tossed. Find the Variance X, i.e., . Solution We saw in Example 2.1 above that X 2 3 4 5 6 7 8 9 10 11 12 f(x) = P(X=x) 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36 Mean of X = E (x) = 7 (see Example 2.1 above) Evaluating E (x2), we get Properties of Variance Let X and Y be random variables and c any real constant. Then (i) Var (c) = 0 (ii) Var (cX) = c2Var (X) (iii) If X and Y are not independent, Var (X + Y) = Var (X) + Var (Y) + 2 Cov (X, Y) where Cov (X, Y) = covariance of X and Y