Question

The density function ofXis given by fX(x) ={a+bx^2if 0≤x≤1 0 otherwise}.If E(X) = 3/5, find a and b.

Answer 1

Answer:

Given,

To determine the a and b values

let us consider expectation of x or mean

i.e.,

E(x) = ∫ x.f(x) dx ---------->(1)

E(x) = 3/5

substitute mean = E(X) = 3.5 in (1)

E(x) = ∫ x*f(x) dx  =3/5

E(x) = ∫ x*(a+bx²) dx = 3/5

split the integration to both individual parts

E(x) = ∫(ax) dx + ∫(bx²) dx = 3/5

E(x) = a∫x dx + b∫x² dx = 3/5

Now by applying the upper limit and lower limit we get

E(x) = a[x²/2]¹₀ + b[x³/3]₀¹ = 3/5

E(x) = (a/2) + (b/4) =3 / 5

10 a+5 b = 3 / 5‑‑‑‑‑‑‑‑ (2)

let us assume,

∫ f(x) dx = 1

substitute the given density function in f(x)

∫ (a + bx²) dx = 1

∫ a dx+ ∫ (bx²) dx   = 1

a∫ dx+ b∫ x² dx   = 1

after the substitution

a+ (b/3) = 1

3a+b = 3 ‑‑‑‑‑‑‑‑‑‑‑‑(3)

Now by solving 2 and 3 equations

we get the a b values as follows

i.e.,

a = 3 / 5 = 0.6

b = 6 / 5 = 1.2