To find $P(X \leq 3)$ for the given discrete probability distribution, we need to sum the probabilities for all values of $X$ that are less than or equal to 3.

Given:

• $P(X = 1) = \frac{1}{5}$
• $P(X = 2) = \frac{2}{5}$
• $P(X = 3) = \frac{1}{5}$
• $P(X = 4) = \frac{1}{5}$

Step 1: Identify the relevant probabilities.

We need to find the probability that $X$ takes a value of 1, 2, or 3: $P(X \leq 3) = P(X = 1) + P(X = 2) + P(X = 3)$

Step 2: Substitute the given probabilities.

$P(X \leq 3) = \frac{1}{5} + \frac{2}{5} + \frac{1}{5}$

Step 3: Add the probabilities.

$P(X \leq 3) = \frac{1}{5} + \frac{2}{5} + \frac{1}{5} = \frac{4}{5}$

Conclusion:

$P(X \leq 3) = \frac{4}{5}$

Would you like more details or have any questions? Here are some related questions:

1. What is a discrete probability distribution?
2. How do you calculate the cumulative probability for a discrete random variable?
3. What is the difference between cumulative distribution function (CDF) and probability mass function (PMF)?
4. How do you verify if a discrete probability distribution is valid?
5. What is the expected value for a discrete random variable?
6. How is the variance of a discrete random variable calculated?
7. Can you explain the concept of independence for discrete random variables?
8. How do you interpret the cumulative probability $P(X \leq x)$?

Tip: When working with discrete probability distributions, the cumulative probability $P(X \leq x)$ is simply the sum of probabilities for all values of $X$ up to and including $x$.