Final Answer:
The value of c in the density function f(y) = ∫ᶜ⁽²−ʸ⁾₀, 0 ≤ y ≤ 2 is 2.
Step-by-step explanation:
Density Function Property: A density function integrates to 1 over its defined domain. Therefore, ∫f(y) dy = 1 within the given interval.
Substituting the Function: Applying this principle to f(y), we get:
∫₀² ∫ᶜ⁽²−ʸ⁾₀ dy = 1
Solving for c: Solving the inner integral first will involve integrating 2 - y from c to 0. To make the outer integral easier to solve, we need both inner and outer integrals to have the same upper bound. Therefore, we set c = 2, making the inner integral:
∫₀² (2 - y) dy = 4 - 2y²/2 |₀² = 4
Substituting the Inner Integral Result: Plugging this result back into the outer integral and simplifying:
∫₀² 4 dy = 4y |₀² = 8
Normalizing and Finding c: For the density function to integrate to 1, we need to divide by 8:
∫₀² ∫ᶜ⁽²−ʸ⁾₀ dy / 8 = 1
=> ∫ᶜ⁽²−ʸ⁾₀ = 8
=> c = 2
Therefore, the value of c that makes the density function integrate to 1 within its domain is 2.