Question

The probability distribution of random variable X is f(x) = ksin(πx/5)
, 0 ≤ x ≤ 5. Determine
the constant k. Also check whether the given function satisfies the conditions of being a
probability density function.

Answer 1

Since f(x) is a p.d.f. ∫−∞∞f(x)dx=1

⇒∫01k(1−x2)dx=1

⇒k[x−3x3]01=1

⇒k(1−31)=1

⇒32k=1

⇒k=23

(ii) The distribution function F(x)=∫−∞xf(t)dt

(a) When x∈(−∞,0]

F(x)=∫−∞xf(t)dt=0

(b) When xε(0,1]

F(x)=∫−∞xf(t)dt=∫−∞0f(t)dt+∫0xf(t)dt

=0+23∫0x(1−t2)dt

F(x)=23(x−3x3)

(c) When x∈[1,∞)

F(x)=∫−∞xf(t)dt

=∫−∞0f(t)dt+∫01f(t)dt+∫1xf(t)dt

=0+∫0123(1−t2)dt+0

=23[t3t3]01=1

∴F(x)=⎩⎪⎪⎪⎨⎪⎪⎪⎧023(x−3x3),1−∞<x≤00<x<11≤x∞[/tex]

✍Hope it's helpful to you ✍