Statistics for Data Science II (BSMA1004) PYQs

Suppose $X \sim \text{Bernoulli}(1/3)$, $Y \sim \text{Bernoulli}(1/2)$, and $Z \sim \text{Bernoulli}(2/3)$ such that $f_{XYZ}(0, 0, 1) = \frac{1}{6}$ and $f_{ZY=1}(0) = \frac{1}{3}$. Find the value of $f_{X|Y=0,Z=1}(0)$.

A fair six-sided die is marked '1' on one face, '2' on two of its faces, and '3' on the remaining three faces. The die is thrown twice. Let $X$ denote the number obtained on the first throw and $Y$ denote the number obtained on the second throw. Define $Z = X + Y$. Find the range of $Z$.

An unbiased coin is tossed three times independently. Let $X$ represent the number of heads in all three tosses, $Y$ represent the number of tails in the first two tosses. Let $U = XY$. Find the PMF of $U$.