List of top Statistics Questions on Distribution Theory

Let X and Y be i.i.d. binomial random variables with parameters n and 0.5 and let Z be another binomial random variable with parameters 2n and 0.5. Then, \(P(X = Y)\) equals to:

Let \(X\) and \(Y\) be independently distributed Poisson random variables such that \(P[X=1]=P[X=2]\), \(P[Y=3]=P[Y=4]\). Then the variance of \((2X-Y)\) is:

The random variable \(X\) represents the number of girls in a family of three children. Assuming that boys and girls are equally likely, what are the mean and standard deviation for the random variable \(X\)?

Three functions \(F_1(x)\), \(F_2(x)\) and \(F_3(x)\) are defined below:

(i) \(F_1(x) = 0\), for all \(x \in (-\infty, +\infty)\)
(ii) \(F_2(x) = 1\), for all \(x \in (-\infty, +\infty)\)
(iii) \(F_3(x) = 0\), for all \(x \le 0\) and \(F_3(x) = 1\), for all \(x > 0\)

Which of the above functions is a distribution function of a random variable?

For a uniform distribution in the range \([0, k]\), the mean and the variance are equal if: