1. For a random variable X~ N(u, o2), derive the following results. (a) (5 points) E[X]...

 

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1. For a random variable X~ N(u, o2), derive the following results. (a) (5 points) E[X] = μ (b) (5 points) Var[X] = 0² (Note that these are not definitions. You must derive them using the functional form of the Normal PDF and integrating it over the measurement of X.) 2. (10 points) For discrete random variables X and Y with joint probabilities of P(X= x, Y = y) for all a X and y Ey, prove that the following holds: Hint: Ey [Y] = Ex[Ey [Y|X]]. P(Y = y|X = x) = P(Y=y, X = x) P(X = x) 1. For a random variable X~ N(u, o2), derive the following results. (a) (5 points) E[X] = μ (b) (5 points) Var[X] = 0² (Note that these are not definitions. You must derive them using the functional form of the Normal PDF and integrating it over the measurement of X.) 2. (10 points) For discrete random variables X and Y with joint probabilities of P(X= x, Y = y) for all a X and y Ey, prove that the following holds: Hint: Ey [Y] = Ex[Ey [Y|X]]. P(Y = y|X = x) = P(Y=y, X = x) P(X = x)