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 for the random variable \( X \):

\[
\begin{array}{c|c|c|c|c|c}
X & 0 & 1 & 2 & 3 & 4 \\
\hline
P(X = x) & K & 2K & 3K & 5K & 2K \\
\end{array}
\]

### (i) Find the value of \( K \)

The sum of the probabilities for all possible values of \( X \) must equal 1. Therefore, we can write the equation:

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

Simplifying the left side:

\[
13K = 1
\]

Now, solve for \( K \):

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

### (ii) Find \( P(X \geq 2) \), \( P(X < 2) \), and \( P(X \leq 3) \)

1. **\( P(X \geq 2) \):** This is the probability that \( X \) is 2, 3, or 4. So:

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

Substituting the values:

\[
P(X \geq 2) = 3K + 5K + 2K = 10K
\]

Using \( K = \frac{1}{13} \):

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

2. **\( P(X < 2) \):** This is the probability that \( X \) is 0 or 1. So:

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

Substituting the values:

\[
P(X < 2) = K + 2K = 3K
\]

Using \( K = \frac{1}{13} \):

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

3. **\( P(X \leq 3) \):** This is the probability that \( X \) is 0, 1, 2, or 3. So:

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

Substituting the values:

\[
P(X \leq 3) = K + 2K + 3K + 5K = 11K
\]

Using \( K = \frac{1}{13} \):

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

### Summary of Results:
- \( K = \frac{1}{13} \)
- \( P(X \geq 2) = \frac{10}{13} \)
- \( P(X < 2) = \frac{3}{13} \)
- \( P(X \leq 3) = \frac{11}{13} \)

Would you like more details or have any questions?

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**Relative Questions:**
1. What is the probability \( P(X = 3) \)?
2. How do you interpret the cumulative distribution function (CDF) for this distribution?
3. What is the probability that \( X \) is greater than 1 but less than 4?
4. Can you find the expected value \( E(X) \) of this distribution?
5. How would the probabilities change if \( K = \frac{1}{10} \)?

**Tip:** Always check that the sum of all probabilities equals 1 to ensure that the distribution is valid.

Explore the probability distribution of random variable X with detailed calculations. 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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