Show that the characteristic function of a Gaussian random variable with zero mean and variance o2 is Ox (@) = exp -0² w² 2
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- If Y is an exponential random variable with parameters beta then mean = E(Y) =beta and variance squaared = V(Y) = beta squared. Show proof of thisShow that the characteristic function of a Gaussian random variable with zero mean and variance o² is (= x (@) = exp -0² w² 2Show that variance o? = (x²) – ((x))²
- Suppose X and Y are independent. X has a mean of 1 and variance of 1, Y has a mean of 0, and variance of 2. Let S=X+Y, calculate E(S) and Var(S). Let Z=2Y^2+1/2 X+1 calculate E(Z). Hint: for any random variable X, we have Var(X)=E(X-E(X))^2=E(X^2 )-(E(X))^2, you may want to find EY^2 with this. Calculate cov(S,X). Hint: similarly, we have cov(Z,X)=E(ZX)-E(Z)E(X), Calculate cov(Z,X). Are Z and X independent? Are Z and Y independent? Why? What about mean independence?let x be a random variable with moment generating function Mx(t)=(0.6 + 0.4e^t)^20 then the variance of x is