Question

Suppose X is a random variable from an exponential family distribution with unknown parameter θ where T(X)= ˣ²η(θ)=−

1/ θA(η)=−log(−η)−log(2), and h(x)=x. - Find E ˣ²and Var ˣ²

Answer 1

To find E and Var of ˣ² from an exponential family distribution with unknown parameter θ, we use the properties of the distribution. E(ˣ²) = 2/θ² and Var(ˣ²) = 2/θ².

Step-by-step explanation:

To find the mean (E) and variance (Var) of ˣ², we need to use the properties of the exponential family distribution.

1. Mean (E): We can use the property ˣ² = Var(X) + (E(X))². Since T(X) = ˣ², we know that Var(X) = E(ˣ²) - (E(X))². From the given information, we have E(X) = η'(θ) = 1/θ and Var(X) = A''(η) = 1/θ². Therefore, E(ˣ²) = Var(X) + (E(X))² = 1/θ² + (1/θ)² = (2/θ²).

2. Variance (Var): From the given information, we already have Var(X) = A''(η) = 1/θ². Therefore, Var(ˣ²) = 2Var(X) = (2/θ²).