Question

Let U₁, U₂ be i.i.d. Unif(0,1). U₁ + U₂ has a Triangle Distribution, with PDF given by

{t for 0 < t ≤ 1,
f(t) =
{2 - t for 1 < t < 2
Show that U₁ + U₂ has a Triangle Distribution by showing that they have the same MGF.

Answer 1

To show that U₁ + U₂ has a Triangle Distribution, we need to show that it has the same Moment Generating Function (MGF) as the Triangle Distribution. The MGF of the Triangle Distribution is (1 - e^(-t))/(t^2). To find the MGF of U₁ + U₂, we find the MGF of U₁ and U₂ and multiply them together.

Step-by-step explanation:

To show that U₁ + U₂ has a Triangle Distribution, we need to show that it has the same Moment Generating Function (MGF) as the Triangle Distribution.

The MGF of the Triangle Distribution is given by M(t) = (1 - e^(-t))/(t^2), where t is a real number.

To find the MGF of U₁ + U₂, we need to find the MGF of the sum of the independent random variables U₁ and U₂. Since U₁ and U₂ are i.i.d. Unif(0,1), their MGFs are given by M₁(t) = e^t/(t+1) and M₂(t) = e^t/(t+1), respectively.

Using the property that the MGF of the sum of independent random variables is the product of their individual MGFs, we have:

M(t) = M₁(t) * M₂(t) = (e^t/(t+1)) * (e^t/(t+1)) = e^(2t)/((t+1)^2)