Questions & Answers of Random Variables

Let X be a Gaussian random variable with mean 0 and variance $\style{font-family:'Times New Roman'}{\sigma^2}$. Let Y= max(X,0) where max(a,b) is the maximum of a and b. The median of Y is ___________.

For any discrete random variable X, with probability mass function

$\style{font-family:'Times New Roman'}{P(X=j)=p_j,p_j\geq0,\;j\in\{0,.....,N\},\;and\;\sum_\limits{j=0}^Np_j=1}$ , define the polynomial function $\style{font-family:'Times New Roman'}{g_X\left(z\right)=\sum_\limits{j=0}^Np_jz^j}$. For a certain discrete random variable Y, there exist a scalar $\style{font-family:'Times New Roman'}{\beta\in\left[0,1\right]}$ such that $\style{font-family:'Times New Roman'}{g_Y=(1-\beta+\beta\;z)^N}$. The expectation of Y is 

Four fair six-sided dice are rolled. The probability that the sum of the results being 22 is X1296. The value of X is _________.

Consider a random variable X that takes values +1 and −1 with probability 0.5 each. The values of the cumulative distribution function F(x) at x = −1 and +1 are