Please solve this

Question

Suppose that X is a continuous random variable whose probability density function is given by f (x) =C(4x −2x2), 0<x <2 0, (1) What is the value of C? (2) Find P(0 < X < 1). (3) Find P(X > 1).

Answer 1

X is a continuous random variable whose probability density function is given by f(x) = C⋅(4x −2x²), 0 < x < 20

1. What is the value of C?

20

∫ f(x) dx = 1

0

20

⇒ ∫ C⋅(4x-2x²) dx = 1

0

⇒ 800-16000/3 = 1/C

⇒ C = -3/13600

2. Find P(0 < X < 1)

P(0 < X < 1)

1

= ∫ f(x) dx

0

1

= ∫ C⋅(4x-2x²) dx

0

= C⋅(2-2/3)

= 4C/3

3. Find P(X > 1)

P(X > 1)

20

= ∫ f(x) dx

1