This question has 4 parts that are dependent on each other. If you could please

Question

This question has 4 parts that are dependent on each other. If you could please show the work that would be phenominal!
Let X and Y be two independent Gaussian random variables with equal mean and variance and 2 respectively.
a) Calculate the expectation of W = u(X 1) where u(x) is the step function.
b) Calculate the CDF of Z = g(X,Y) -->
c) Assume that Xi, for i = 1, . . . , n are independent
variables and let Zn =p(X1 +X2 .... +Xn ). What is the probability that Zn >1 for n
d) Calculate the conditional expectation of Z3 given Y = .

Explanation / Answer

multiple questions:

a) step fucntion - u(x) = 0 (if x<0) / 1 (if(x>=0)

so, u(x-1) = 0 (if x<1) / 1 (if(x>=1)

expectation of x = E(x) = sigma (each of possible outcomes * probability of outcome of (X))

i.e E(X) = Sigma( S * P(X))

E(X)= 0 * 1/2 + 1* 1/2 =1/2 ( for step function)

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