Question: User: An insurance company offers its policyholders a number of different premium payment options. For a randomly selected policyholder, let $X$ be the number of months between successive payments. The cumulative distribution function of $X$ is given.
(a) What is the probability mass function of $X$?
(b) Compute $P(2 < X \leq 7)$.
$$F(x) = \begin{cases} 0, & \text{if } x < 1, \\ 0.2, & \text{if } 1 \leq x < 3, \\ 0.5, & \text{if } 3 \leq x < 5, \\ 0.8, & \text{if } 5 \leq x < 7, \\ 1, & \text{if } x \geq 7. \end{cases}$$
(a) What is the probability mass function of $X$?
$x \quad 1 \quad 3 \quad 5 \quad 7$
$f(x)$
1. **State the problem:** We are given the cumulative distribution function (CDF) $F(x)$ of a discrete random variable $X$ representing the number of months between successive payments. We need to find:
(a) The probability mass function (PMF) of $X$.
(b) The probability $P(2 < X \leq 7)$.
2. **Recall the relationship between CDF and PMF:**
For a discrete random variable, the PMF $f(x)$ is the probability that $X$ takes the value $x$. It can be found from the CDF $F(x)$ by calculating the jump sizes at each point:
$$f(x) = P(X = x) = F(x) - \lim_{t \to x^-} F(t)$$
3. **Identify the jump points from the CDF:**
The CDF jumps at $x = 1, 3, 5, 7$.
4. **Calculate the PMF values:**
- At $x=1$:
$$f(1) = F(1) - F(1^-) = 0.2 - 0 = 0.2$$
- At $x=3$:
$$f(3) = F(3) - F(3^-) = 0.5 - 0.2 = 0.3$$
- At $x=5$:
$$f(5) = F(5) - F(5^-) = 0.8 - 0.5 = 0.3$$
- At $x=7$:
$$f(7) = F(7) - F(7^-) = 1 - 0.8 = 0.2$$
5. **Summarize the PMF:**
$$\begin{array}{c|cccc}
x & 1 & 3 & 5 & 7 \\
f(x) & 0.2 & 0.3 & 0.3 & 0.2 \\ \end{array}$$
6. **Compute $P(2 < X \leq 7)$:**
Since $X$ is discrete and takes values only at $1,3,5,7$, the event $2 < X \leq 7$ includes $X=3,5,7$.
$$P(2 < X \leq 7) = f(3) + f(5) + f(7) = 0.3 + 0.3 + 0.2 = 0.8$$
7. **Final answers:**
- PMF: $f(1)=0.2$, $f(3)=0.3$, $f(5)=0.3$, $f(7)=0.2$
- $P(2 < X \leq 7) = 0.8$