Answer:
Explanation:
To find the moment-generating function (MGF) of a random variable Y, we need to follow these steps:
1. Start with the probability density function (PDF) of Y, which is given as:
f(y) = 8y^3e^(-2y) for y > 0
2. Recall that the MGF of Y is defined as the expected value of e^(tY), where t is a constant:
M(t) = E[e^(tY)]
3. To find the MGF, we need to calculate the expected value by integrating e^(ty) multiplied by the PDF:
M(t) = ∫(e^(ty) * f(y)) dy
4. Substitute the given PDF into the integral:
M(t) = ∫(e^(ty) * 8y^3e^(-2y)) dy
5. Simplify the integral:
M(t) = 8 ∫(y^3e^((ty-2y))) dy
= 8 ∫(y^3e^(-y(2-t))) dy
6. Now, let's focus on the exponent term inside the integral. We can rewrite it as:
-y(2-t) = -y*2 + yt
7. Rewrite the integral:
M(t) = 8 ∫(y^3e^(-y*2) * e^(yt)) dy
8. Notice that the integral ∫(y^3e^(-y*2)) represents the gamma function, which has the property:
∫(y^ne^(-ky)) dy = n! / k^(n+1)
9. Using this property, we can simplify the integral:
M(t) = 8 * 3! / (2^4) * ∫(y^3e^(-y*2)) dy
= 24 / 16 * ∫(y^3e^(-y*2)) dy
= 3 / 2 * ∫(y^3e^(-y*2)) dy
10. Finally, the MGF becomes:
M(t) = 3 / 2 * ∫(y^3e^(-y*2)) dy
And that's the final expression for the moment-generating function of Y