Question

The probability density function for a random variable X is given by f(x) = x 18 , 0 < X < 6. Use this pdf to find the following probabilities. (Hint: draw the pdf and remember that probability = area.) a. Find P(X < 1). (4 points) b. Find P(X > 4). (6 poin

Answer 1

a) 0.0278

b) 0.5556

Explanation:

We are given the following in the question:

The probability density function for a random variable X is given by

`f(x) = \displaystyle(x)/(18)\\\\0 < x < 6`

We can find the probabilities as:

`P(X<c) = \displaystyle\int_(-\infty)^c f(x) dx\\\\\\P(X>c) = \displaystyle\int^(\infty)_c f(x) dx`

a) P(X < 1)

`P(X<1) = \displaystyle\int_(-\infty)^1 f(x) dx\\\\P(X<1) =\displaystyle\int_(0)^1 (x)/(18) dx\\\\= (1)/(18)\displaystyle\int_(0)^1 x~ dx = (1)/(18)\Big[(x^2)/(2)\Big]^1_0\\\\= (1)/(18)\Big((1)/(2)-0\Big) = (1)/(36) = 0.0278`

b) P(X > 4)

`P(X>4) = \displaystyle\int^(\infty)_4 f(x) dx\\\\P(X>4) = \displaystyle\int^(6)_4 (x)/(18) dx\\\\= (1)/(18)\displaystyle\int_(4)^6 x~ dx = (1)/(18)\Big[(x^2)/(2)\Big]^6_4\\\\= (1)/(18)\Big((36)/(2)-(16)/(2)\Big) = (1)/(36)\Big(10\Big) = (10)/(18) = 0.5556`