Question

The probability distribution of a discrete random variable X is given by

P(X = x) = x²/14, x ∈ {1, 2, k}, where k > 0.
(a) Use the formula to calculate E[X], the expected value of X.
(b) Calculate E[X²] and hence the variance σ² = Var(X) and standard deviation σ.

Answer 1

The expected value of a discrete random variable X is calculated by multiplying each value of the random variable by its probability and adding the products. The variance can be found by subtracting the square of the expected value from the expected value of X². The standard deviation is the square root of the variance.

Step-by-step explanation:

To calculate the expected value, E[X], of a discrete random variable X, we need to multiply each value of the random variable by its corresponding probability and add the products. Given the probability distribution P(X = x) = x²/14, where x ∈ {1, 2, k}, we have:

E[X] = Σ xP(x)

= (1 * 1²/14) + (2 * 2²/14) + (k * k²/14)

= 1/14 + 8/14 + k³/14

Therefore, the expected value of X is E[X] = 9/14 + k³/14.

To calculate E[X²], we need to square each value of the random variable, multiply by its corresponding probability, and add the products. Using the same probability distribution, we have:

E[X²] = Σ x²P(x)

= (1² * 1²/14) + (2² * 2²/14) + (k² * k²/14)

= 1/14 + 4/14 + k⁴/14

Hence, E[X²] = 5/14 + k⁴/14.

The variance, σ², can be calculated as Var(X) = E[X²] - (E[X])². Therefore, the variance of X is σ² = (5/14 + k⁴/14) - (9/14 + k³/14)².

To calculate the standard deviation, σ, we can simply take the square root of the variance σ². Hence, the standard deviation of X is σ = √((5/14 + k⁴/14) - (9/14 + k³/14)²).