A random variable T~ gamma (a,^) has density function: (^t)a-1, + >o a) show that it belongs to an exponential family. Define neutral parameters as functions of a and λ. -1t ле fy(t) = f(a) 10² гса) b) let X,~ Gamma (a, 1) and X₂ ~ Gamma (agil) be independent random variables. Let Y₁ = x, and Y₂ = X₁ + X₂ Find the marginal probability density functions for Y₁ and 1₂ Are they independent? ex/

Step 1: Write the given information

Step 2: Determine whether the given pdf belongs to an exponential family and define neutral parameters

Step 3: Find the marginal probability density functions for Y1 and Y2 and check that are they independent

Solution