Question

Let X₁, X₂ two independent and identically distribution geometric random variables with common pmf f(k) = P(X₁ = k) = p(1 - p)ᵏ for k = 0,1,.... Pay attention to the fange.

a) Directly compute P(X₁ + X₂ = k) for k = 0,1,....
b) Derive the MGF of X₁ + X₂ based on the result obtained in a).
c) Derive the MGF of X₁+ X₂ based on MGF of X₁ and X₂

Answer 1

To compute the probability that the sum of two independent and identically distributed geometric random variables X₁ and X₂ equals a given value k, use the convolution formula. The MGF of X₁ + X₂ can be derived by using the result from part a). The MGF of X₁ + X₂ can also be derived by using the MGFs of X₁ and X₂.

Step-by-step explanation:

To compute the probability that the sum of two independent and identically distributed (i.i.d.) geometric random variables X₁ and X₂ equals a given value k, we need to consider all the possible values of X₁ and X₂ that add up to k. We can do this by using the convolution formula, which states that P(X₁ + X₂ = k) = ∑ P(X₁ = i) * P(X₂ = k-i), where the summation is over all possible values of i from 0 to k. In this case, the pmf of X₁ and X₂ is given as f(k) = p(1 - p)ᵏ for k = 0, 1, ..., where p is the probability of success in a single trial.

To derive the moment generating function (MGF) of X₁ + X₂, we can use the result obtained in part a). The MGF of a sum of independent random variables is equal to the product of their individual MGFs. Since X₁ and X₂ are i.i.d., their MGFs are the same. Therefore, the MGF of X₁ + X₂ is equal to the square of the MGF of X₁.

To derive the MGF of X₁ + X₂ based on the MGFs of X₁ and X₂, we can use the fact that the MGF of a sum of independent random variables is equal to the product of their individual MGFs. Therefore, the MGF of X₁ + X₂ is equal to the product of the MGFs of X₁ and X₂.