Question

Given that probability density function of a continuous random variable x, as
f (x) = $ - \dfrac{{{x^2}}}{3}$, -1 < x < 2
      = 0, otherwise
Then P(x > 0)
$\left( a \right)\dfrac{2}{9}$
$\left( b \right)\dfrac{1}{9}$
$\left( c \right)\dfrac{8}{9}$
$\left( d \right)\dfrac{4}{9}$

Answer 1

Hint: In this particular type of question use the concept of finding the probability under the limits (a < x < b) of probability density function is $\int\limits_a^b {f\left( x \right)} dx$, and use the basic integration formula to integrate it so use these concepts to reach the solution of the question.

Complete step by step answer:
Given function:
f (x) = $ - \dfrac{{{x^2}}}{3}$, -1 < x < 2
      = 0, otherwise
Where f(x) is a probability density function.
Now probability of given probability density function f (x) under the limits a < x < b is given as,
P (a < x < b) = $\int\limits_a^b {f\left( x \right)} dx$
Now we have to find the probability for (x > 0)
Now as it is given that the probability density function f (x) is defined between the interval (-1, 2) otherwise it is zero.
So we have to find the probability for (x > 0), therefore, we have to take the integral limits from 0 to 2.
So the probability of the given probability density function is,
P (x > 0) = $\int_0^2 {f\left( x \right)dx} $
Now substitute the value of f (x) we have,
$ \Rightarrow $P (x > 0) = $\int_0^2 {\dfrac{{ - {x^2}}}{3}dx} $
Now as we know that $\int {{x^n}dx = \dfrac{{{x^{n + 1}}}}{{n + 1}} + c} $, where c is some arbitrary integration constant.
So use this property in the above integral we have,
$ \Rightarrow $P (x > 0) = $ - \dfrac{1}{3}\left[ {\dfrac{{{x^3}}}{3}} \right]_0^2$
Now apply integral limits we have,
$ \Rightarrow $P (x > 0) = $ - \dfrac{1}{3}\left[ {\dfrac{{{2^3}}}{3} - 0} \right]$
Now simplify this we have,
$ \Rightarrow $P (x > 0) = $ - \dfrac{1}{3}\left[ {\dfrac{8}{3}} \right] = \dfrac{{ - 8}}{9}$
So this is the required answer.
Hence option (C) is the correct answer.


Note: Whenever we face such types of questions the key concept we have to remember is that always recall the basic formula of integration such as, $\int {{x^n}dx = \dfrac{{{x^{n + 1}}}}{{n + 1}} + c} $, so first use the probability formula of finding the probability under the given limits then apply this basic integration formula as above and simplify we will get the required answer.