Solution:

Given probability distribution of random variable X is as follows:

X: 1 2 4 6 8
P: k 2k 3k 3k k

To find the variance of X, we need to find the expected value of X and expected value of X squared.

Finding Expected Value of X:

The expected value of X is given by the formula:

E(X) = Σ Xi * Pi

where Xi is the value of the random variable and Pi is the corresponding probability.

So, E(X) = 1k + 2(2k) + 4(3k) + 6(3k) + 8k
E(X) = k + 4k + 12k + 18k + 8k
E(X) = 43k

Finding Expected Value of X Squared:

The expected value of X squared is given by the formula:

E(X²) = Σ Xi² * Pi

where Xi is the value of the random variable and Pi is the corresponding probability.

So, E(X²) = 1²k + 2²(2k) + 4²(3k) + 6²(3k) + 8²k
E(X²) = k + 8k + 48k + 108k + 64k
E(X²) = 229k

Finding Variance of X:

The variance of X is given by the formula:

Var(X) = E(X²) - [E(X)]²

So, Var(X) = 229k - (43k)²
Var(X) = 229k - 1849k²

Explanation:

- The probability distribution of a random variable gives the probabilities of all possible values of the random variable.
- The expected value of a random variable is the sum of the products of the values of the random variable and their corresponding probabilities.
- The expected value of a random variable squared is the sum of the products of the squares of the values of the random variable and their corresponding probabilities.
- The variance of a random variable is a measure of how far the values of the random variable are spread out from their expected value.
- The variance of a random variable is the expected value of the squared deviations of the values of the random variable from their expected value.

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