We are given a random variable $X$ with the following probability distribution:

$X = 0, 1, 2, 3, 4, 5$ $P(X) = 0.1, K, 0.2, 2K, 0.3, K$

i. To find $K$:

The sum of all probabilities must equal 1. Therefore, we can write the equation:

$0.1 + K + 0.2 + 2K + 0.3 + K = 1$

Simplifying:

$0.1 + 0.2 + 0.3 + K + 2K + K = 1$ $0.6 + 4K = 1$

Now solve for $K$:

$4K = 1 - 0.6$ $4K = 0.4$ $K = \frac{0.4}{4} = 0.1$

ii. To find $P(X \leq 1)$:

To find $P(X \leq 1)$, we sum the probabilities for $X = 0$ and $X = 1$:

$P(X \leq 1) = P(X = 0) + P(X = 1)$ $P(X \leq 1) = 0.1 + K = 0.1 + 0.1 = 0.2$

iii. To find $P(X > 3)$:

To find $P(X > 3)$, we sum the probabilities for $X = 4$ and $X = 5$:

$P(X > 3) = P(X = 4) + P(X = 5)$ $P(X > 3) = 0.3 + K = 0.3 + 0.1 = 0.4$

Summary of answers:

• i. $K = 0.1$
• ii. $P(X \leq 1) = 0.2$
• iii. $P(X > 3) = 0.4$

Would you like further explanation on any part of the solution?

Here are some related questions to explore:

1. How do we calculate the expected value $E(X)$ of a discrete random variable?
2. What is the formula to find the variance $\text{Var}(X)$ of a discrete random variable?
3. How would you find the cumulative distribution function (CDF) from a probability distribution?
4. What is the difference between a probability mass function (PMF) and a cumulative distribution function (CDF)?
5. Can you explain the concept of conditional probability using a probability distribution?

Tip: Always check that the total probability in any probability distribution sums to 1. If it doesn't, something is wrong with the given probabilities.