1. **Problem Statement:** Show that the function $P(X) = \frac{X}{10}$ for $X=1,2,3,4$ is a probability mass function (pmf) of a discrete random variable $X$. Then find: i. $P(X \leq 2)$ ii. $P(X > 2)$ iii. $P(1 < X < 3)$ 2. **Definition and Rules:** A function $P(X)$ is a pmf if: - $P(X) \geq 0$ for all $X$ - $\sum P(X) = 1$ over all possible values of $X$ 3. **Check if $P(X)$ is a pmf:** Calculate $P(1), P(2), P(3), P(4)$: $$P(1) = \frac{1}{10} = 0.1$$ $$P(2) = \frac{2}{10} = 0.2$$ $$P(3) = \frac{3}{10} = 0.3$$ $$P(4) = \frac{4}{10} = 0.4$$ Sum: $$0.1 + 0.2 + 0.3 + 0.4 = 1.0$$ All probabilities are non-negative and sum to 1, so $P(X)$ is a valid pmf. 4. **Calculate requested probabilities:** i. $P(X \leq 2) = P(1) + P(2) = 0.1 + 0.2 = 0.3$ ii. $P(X > 2) = P(3) + P(4) = 0.3 + 0.4 = 0.7$ iii. $P(1 < X < 3)$ means $X=2$ only, so: $$P(2) = 0.2$$ **Final answers:** - $P(X \leq 2) = 0.3$ - $P(X > 2) = 0.7$ - $P(1 < X < 3) = 0.2$