Question
Problem(1) Given a continuous random variable X with probability density function fX (x) = f(x), say, x can be any real number. (a) What is
Problem(1) Given a continuous random variable X with probability density function
fX (x) = f(x), say, x can be any real number.
(a) What is the definition of the expectation of the random variable X, i.e. E(X) =?
(b) If we denote E(X) by X , then the definition of the variance of the random
variable X is defined by Var(X) = E[(X X )2]; show that Var(X) = E(X2) 2
X .
(c) If we have another continuous random variable Y (with expectation Y ), the covariance of the two random variables X and Y is defined by Cov(X, Y ) = E[(X X )(Y Y )] . Show that Cov(X, Y ) = E(XY ) E(X)E(Y ). (That is, Cov(X, Y ) = E(XY ) X Y .)
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