Let the probability distribution of X be given by X 0 1 2 3 4 . P(X=x) K 2K 3K 5K 2K  .i) Find the value of K  .ii) Find P(x≥2), P(x<2), P(x≤3)

Given the probability distribution of \( X \) as follows:

\[
X: \quad 0 \quad 1 \quad 2 \quad 3 \quad 4
\]
\[
P(X = x): \quad K \quad 2K \quad 3K \quad 5K \quad 2K
\]

### i) Finding the value of \( K \):

The sum of all probabilities for a discrete random variable must equal 1:

\[
K + 2K + 3K + 5K + 2K = 1
\]

Simplifying this:

\[
K(1 + 2 + 3 + 5 + 2) = 1
\]

\[
K(13) = 1
\]

\[
K = \frac{1}{13}
\]

### ii) Finding the probabilities:

1. **\( P(X \geq 2) \)**: This includes \( X = 2 \), \( X = 3 \), and \( X = 4 \).

\[
P(X \geq 2) = P(X = 2) + P(X = 3) + P(X = 4)
\]

\[
P(X \geq 2) = 3K + 5K + 2K = 10K = 10 \times \frac{1}{13} = \frac{10}{13}
\]

2. **\( P(X < 2) \)**: This includes \( X = 0 \) and \( X = 1 \).

\[
P(X < 2) = P(X = 0) + P(X = 1)
\]

\[
P(X < 2) = K + 2K = 3K = 3 \times \frac{1}{13} = \frac{3}{13}
\]

3. **\( P(X \leq 3) \)**: This includes \( X = 0 \), \( X = 1 \), \( X = 2 \), and \( X = 3 \).

\[
P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)
\]

\[
P(X \leq 3) = K + 2K + 3K + 5K = 11K = 11 \times \frac{1}{13} = \frac{11}{13}
\]

### Summary:

1. \( K = \frac{1}{13} \)
2. \( P(X \geq 2) = \frac{10}{13} \)
3. \( P(X < 2) = \frac{3}{13} \)
4. \( P(X \leq 3) = \frac{11}{13} \)

Would you like a more detailed explanation or have any questions?

Here are 5 related questions:

1. How would the probability distribution change if \( P(X = 3) \) was doubled?
2. What is the expected value \( E(X) \) for this distribution?
3. How do you calculate the variance of this probability distribution?
4. If another value \( X = 5 \) is added with probability \( K \), how would \( K \) change?
5. What is the cumulative distribution function (CDF) for this random variable \( X \)?

**Tip:** When dealing with probability distributions, always check that the total probability sums to 1 before making any calculations.

Explore the probability distribution of X with step-by-step solutions. Learn how to find the value of K, probabilities for x ≥ 2, x < 2, and x ≤ 3. Suitable for high school students studying probability theory.

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