Hello I need assistance with B24 please I placed another pic

Hello, I need assistance with B.24 please; I placed another picture of A.17 as it\'s needed to complete B.24 thank you very much.

Solution

First we need to find the value of c

Integral of f(X) from 0 to 2 = 1

Integrating f(x) we get

F(x) = c [ 4x² /2 - 2x³/3 ] from 0 to 2 = 1

So, 8/3 c = 1

c = 3/8

Mean = integral of [x f(x) ] from 0 to 2

x f(x) = (3/8) [4x² - 2x³]

Integrating it we get : (3/8) [ 4x³/3 - 2x⁴ / 4]

Substituting the values,

mean = (3/8) * (8/3)

= 1

Variance =

Integral of [ (x - mean)² f(x) ] from 0 to 2

= integral of ( (x-1)² ()4x - 2x² )] * (3/8)

= integral of [ -2x⁴ + 8x³ - 10x² + 4x] * (3/8)

= (3/8) * [ -2x⁵/5 + 2x⁴ - 10x³/3 + 2x² ] from 0 to 2

Thus, variance = 3/8 * [0.533333]

= 1/5

= 0.2

Variance = .0.2

Hope this helps. Ask if you have any doubts.