Given the probability density functio (p.d.f) of a continuos random variable X as. `f(x)=(x^(2))/(3),-1 lt x lt2`
Determine the cumulative distribution function (c.d.f) X and hence find `P(X lt1),P(X gt0), P(1 lt X lt 2)`.

Explanation

c.d.f. of the continuous variable is given by
`f(x) = underset(-1)overset(x) int y^(2)/3 dx `
` = [y^(3)/9]_(-1)^(x)`
` = (((x^(3)+1))/9) , x in R`
Consider ` P (x lt 1) = f(1) = ((1)^(3) + 1)/9 = 2/9`
` P(x le -2) = 0`
` P(x gt 0) = 1 - P (x le 0) `
` = 1 - f(0) = 1 - (1/9) = 8/9`
` P(1 lt x lt 2) = f(2) - f (1) = 1 - 2/9 = 7/9`