Question 1177296
**1. Determine the Relationship Between a and b**

Since f(x) is a probability density function (pdf), the integral of f(x) over its domain must equal 1:

* ∫[0, 1] (ax² + b) dx = 1

Integrate:

* [ (ax³/3) + bx ] from 0 to 1 = 1
* (a/3 + b) - (0) = 1
* a/3 + b = 1
* a + 3b = 3
* a = 3 - 3b

**2. Calculate the Expected Value E(X)**

The expected value of X is given by:

* E(X) = ∫[0, 1] x * f(x) dx
* E(X) = ∫[0, 1] x * (ax² + b) dx
* E(X) = ∫[0, 1] (ax³ + bx) dx

Integrate:

* E(X) = [ (ax⁴/4) + (bx²/2) ] from 0 to 1
* E(X) = (a/4 + b/2) - (0)
* E(X) = a/4 + b/2

**3. Use the Given E(X) = 5/8**

We are given that E(X) = 5/8. So:

* a/4 + b/2 = 5/8

Multiply both sides by 8 to eliminate fractions:

* 2a + 4b = 5

**4. Substitute a = 3 - 3b**

Substitute a = 3 - 3b into the equation 2a + 4b = 5:

* 2(3 - 3b) + 4b = 5
* 6 - 6b + 4b = 5
* 6 - 2b = 5
* -2b = -1
* b = 1/2

**5. Find a**

Substitute b = 1/2 back into a = 3 - 3b:

* a = 3 - 3(1/2)
* a = 3 - 3/2
* a = 6/2 - 3/2
* a = 3/2

**Therefore, a = 3/2 and b = 1/2.**