Question

We determined that

f(y₁, y₂) =

6(1 − y₂), 0 ≤ y₁ ≤ y₂ ≤ 1,
0, elsewhere,

is a valid joint probability density function. It has marginal density functions f₁(y₁) = 3(1 − y₁)², where 0 ≤ y₁ ≤ 1, and f₂(y₂) = 6y₂(1 − y₂),where 0 ≤ y₂ ≤ 1.

(a)Find E(Y₁) and E(Y₂).

Answer 1

To calculate the expected values E(Y₁) and E(Y₂), we integrate the respective marginal density functions over the range from 0 to 1. This process yields the average values of the random variables within their domains.

Step-by-step explanation:

Finding Expected Values E(Y₁) and E(Y₂)

To find the expected values E(Y₁) and E(Y₂) for the continuous random variables Y₁ and Y₂ with given marginal density functions, we use the following formulas:

1. E(Y₁) = ∫ y₁f₁(y₁)dy₁ from 0 to 1
2. E(Y₂) = ∫ y₂f₂(y₂)dy₂ from 0 to 1

For E(Y₁), using the marginal density function f₁(y₁) = 3(1 - y₁)², we integrate:

E(Y₁) = ∫ from 0 to 1 y₁ * 3(1 - y₁)² dy₁

For E(Y₂), using the marginal density function f₂(y₂) = 6y₂(1 - y₂), we integrate:

E(Y₂) = ∫ from 0 to 1 y₂ * 6y₂(1 - y₂) dy₂

These integrations will yield the expected values for Y₁ and Y₂, representing the mean values of these random variables within their respective domains.