Question

Two cards drawn at random from a pack of cards with replacement. Let the random variable x be the number of card drawn from the heart suit.
A. Construct a probability mass function
B. Construct a cumulative distribution function
C. What is the most likely value of the random variable X?

Answer 1

This is a discrete distribution with three possible cases of interest. In any such two-card hand, you can have (1) no cards, (2) one card, or (3) both cards with the heart suit.

$\mathbb P(X=x)=\begin{cases}\dfrac{\dbinom{39}2}{\dbinom{52}2}=\dfrac{19}{34}&\text{for }x=0\\\\\dfrac{\dbinom{39}1\dbinom{13}1}{\dbinom{52}2}=\dfrac{13}{34}&\text{for }x=1\\\\\dfrac{\dbinom{13}2}{\dbinom{52}2}=\dfrac1{17}&\text{for }x=2\end{cases}$

As a PMF, you would say

$f_X(x)=\begin{cases}\dfrac{19}{34}&\text{for }x=0\\\\\dfrac{13}{34}&\text{for }x=1\\\\\dfrac1{17}&\text{for }x=2\\\\0&\text{otherwise}\end{cases}$

The CDF is given by

$\displaystyle F_X(x)=\begin{cases}0&\text{for }x$

The most likely value of $X$ is the expected value for the distribution, which is given by

$\displaystyle\mathbb E(X)=\sum_xxf_X(x)=0\times f_X(0)+1\times f_X(1)+2\times f_X(2)=\dfrac{17}{34}=\dfrac12$