The table provided shows the probability mass function (PMF) $f(x)$ for a discrete random variable. To determine $F(x)$, the cumulative distribution function (CDF), we need to compute the cumulative probabilities up to a specific value of $x$.

Cumulative Distribution Function (CDF)

The CDF, $F(x)$, is defined as: $F(x) = P(X \leq x) = \sum_{k \leq x} f(k)$ In other words, for each value of $x$, $F(x)$ is the sum of all the probabilities for values less than or equal to $x$.

1. Compute $F(4)$

To determine $F(4)$, sum the probabilities for $x = 1, 2, 3, 4$: $F(4) = f(1) + f(2) + f(3) + f(4)$ Using the table: $F(4) = 0.3 + 0.05 + 0.06 + 0.13 = 0.54$ Thus, $F(4) = 0.54$.

2. Compute $F(6) - F(3)$

First, compute $F(6)$ and $F(3)$ and then subtract them.

• $F(6)$ is the cumulative probability for $x \leq 6$: $F(6) = f(1) + f(2) + f(3) + f(4) + f(5) + f(6)$ $F(6) = 0.3 + 0.05 + 0.06 + 0.13 + 0.08 + 0.02 = 0.64$

• $F(3)$ is the cumulative probability for $x \leq 3$: $F(3) = f(1) + f(2) + f(3) = 0.3 + 0.05 + 0.06 = 0.41$

Now subtract $F(3)$ from $F(6)$: $F(6) - F(3) = 0.64 - 0.41 = 0.23$

Summary of Results:

• $F(4) = 0.54$
• $F(6) - F(3) = 0.23$

Would you like more details or further explanation?

Here are five related questions:

1. How do you calculate the CDF for continuous random variables?
2. Can you use this approach for finding probabilities in binomial distributions?
3. What is the relationship between the CDF and PMF for discrete variables?
4. How can the CDF be used to compute expected values of a random variable?
5. How do you compute the variance from the PMF?

Tip: The CDF is non-decreasing and always approaches 1 as $x$ increases for any probability distribution.