A random variable X has the following probability distribution.x 0 1 2 3 4 5 6 7p(x) 0 k 2k 2k 3k k2 2 k2 7k2+kFind (i) The value of k (ii) 𝑃(1.5 < 𝑥 < 4.5/𝑥 > 2) and(iii) The smallest value of n for which 𝑃(𝑋 ≤ 𝑛) > 12.

(i) To find the value of k, we can use the fact that the sum of all probabilities in a probability distribution must equal 1.

Summing up the probabilities given, we have: 0 + k + 2k + 2k + 3k + k^2 + 2k^2 + k^2 + 7k^2 + k = 1

Combining like terms, we get: 15k^2 + 8k + 1 = 1

Simplifying further, we have: 15k^2 + 8k = 0

Factoring out k, we get: k(15k + 8) = 0

Setting each factor equal to zero, we have two possible solutions: k = 0 or 15k + 8 = 0

Since k cannot be zero (as it would make some probabilities zero), we solve for k in the second equation: 15k + 8 = 0 15k = -8 k = -8/15

Therefore, the value of k is -8/15.

(ii) To find P(1.5 < x < 4.5 / x > 2), we need to calculate the probability of the event (1.5 < x < 4.5) given that (x > 2).

First, we need to find the probability of x being between 1.5 and 4.5, which is the sum of the probabilities for x = 2, 3, and 4: P(1.5 < x < 4.5) = P(x = 2) + P(x = 3) + P(x = 4) = 2k + 2k + 3k = 7k

Next, we need to find the probability of x being greater than 2, which is the sum of the probabilities for x = 3, 4, 5, 6, and 7: P(x > 2) = P(x = 3) + P(x = 4) + P(x = 5) + P(x = 6) + P(x = 7) = 2k + 2k + 3k + k^2 + 2k^2 = 7k + k^2 + 2k^2 = 7k + 3k^2

Finally, we can calculate P(1.5 < x < 4.5 / x > 2) by dividing the probability of (1.5 < x < 4.5) by the probability of (x > 2): P(1.5 < x < 4.5 / x > 2) = (7k) / (7k + 3k^2)

(iii) To find the smallest value of n for which P(X ≤ n) > 12, we need to calculate the cumulative probability distribution function (CDF) for X and find the smallest value of n that satisfies the condition.

The CDF for X is the sum of the probabilities for all values of X up to and including n: CDF(n) = P(X ≤ n) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = n)

We need to find the smallest value of n such that CDF(n) > 12.