3

7

5

11. Find P (X ≤ 3) for the following discrete probability distribution function of a

random variable X:

P (X=1) = 1/5

P (X =2) = 2/5

P (X=3) = 1/5

P (X =4) = 1/5

(1 Point)

5

1

5

4

5

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 \( x \).

Learn how to calculate the cumulative probability P(X ≤ 3) for a discrete random variable with a given probability distribution. This problem involves understanding discrete probability distributions and calculating cumulative probabilities. Suitable for Grade 11 students.

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