Question

Find the c.d.f. F(x) associated with the following p.d.f. f(x)

f(x) = `{{:(3(1 - 2x^2)",", 0 < x < 1),(0",", "otherwise"):}`

Find `P(1/4 < x < 1/3)` by using p.d.f. and c.d.f.

Answer 1

c.d.f. of X is given by

∴ F(x) = `int_0^x 3(1 - 2x^2)dx`

= `3[x - (2x^3)/3]_0^x`

= `3((3x - 2x^3))/3`

= 3x – 2x³

Using c.d.f.

`P(1/4 < x < 1/3) = F(1/3) - F(1/4)`

= `(3 xx 1/3 - 2 xx 1/27) - (3 xx 1/4 - 2 xx 1/64)`

= `(1 - 2/27) - (3/4 - 1/32)`

= `25/27 - 23/32`

= `(800 - 621)/864`

= `179/864`