Nov 2010
15. After some rain the depth of moisture, [tex]$X$[/tex] meters, in Arda Gardens can be taken as a continuous random variable with a probability density function [tex]$f(x)=\left\{\begin{array}{ll}\frac{12 x}{5}(b-x), & 0 \leq x \leq 1 \\ 0, & \text { otherwise }\end{array}\right.$[/tex]
a) Find the value of [tex]$b$[/tex].
[3]
b) Calculate the probability that the depth of moisture exceeds 0,9.
[3]
Nov 2014
Answer (1)
Find b by setting the integral of the pdf from 0 to 1 equal to 1: ∫ 0 1 5 12 x ( b − x ) d x = 1 , which gives b = 1.5 .
Calculate 0.9)"> P ( X > 0.9 ) by integrating the pdf from 0.9 to 1: ∫ 0.9 1 5 12 x ( 1.5 − x ) d x .
Evaluate the integral: [ 5 9 x 2 − 5 4 x 3 ] 0.9 1 = 0.1252 .
The probability that the depth of moisture exceeds 0.9 is 0.1252 .
Explanation
Understand the problem and provided data We are given a probability density function (pdf) for the depth of moisture, X , in Arda Gardens:
f ( x ) = { 5 12 x ( b − x ) , 0 ≤ x ≤ 1 0 , otherwise
We need to find the value of b and the probability that the depth of moisture exceeds 0.9, i.e., 0.9)"> P ( X > 0.9 ) .
Find the value of b To find the value of b , we use the property that the integral of the pdf over its entire range must equal 1. This is because the total probability of all possible outcomes must be 1. So, we have:
∫ 0 1 f ( x ) d x = 1
∫ 0 1 5 12 x ( b − x ) d x = 1
Let's solve this integral for b .
Calculate the integral and solve for b ∫ 0 1 5 12 x ( b − x ) d x = ∫ 0 1 ( 5 12 b x − 5 12 x 2 ) d x
= [ 5 6 b x 2 − 5 4 x 3 ] 0 1 = 5 6 b − 5 4 − ( 0 ) = 5 6 b − 4
Now, set this equal to 1:
5 6 b − 4 = 1
6 b − 4 = 5
6 b = 9
b = 6 9 = 2 3 = 1.5
Calculate the probability P(X > 0.9) Now that we have b = 1.5 , we can find the probability that the depth of moisture exceeds 0.9. This is given by:
0.9) = \int_{0.9}^{1} \frac{12x}{5}(1.5 - x) dx = \int_{0.9}^{1} (\frac{18x}{5} - \frac{12x^2}{5}) dx"> P ( X > 0.9 ) = ∫ 0.9 1 5 12 x ( 1.5 − x ) d x = ∫ 0.9 1 ( 5 18 x − 5 12 x 2 ) d x
Let's calculate this integral.
Evaluate the integral ∫ 0.9 1 ( 5 18 x − 5 12 x 2 ) d x = [ 5 9 x 2 − 5 4 x 3 ] 0.9 1
= ( 5 9 ( 1 ) 2 − 5 4 ( 1 ) 3 ) − ( 5 9 ( 0.9 ) 2 − 5 4 ( 0.9 ) 3 )
= ( 5 9 − 5 4 ) − ( 5 9 ( 0.81 ) − 5 4 ( 0.729 ) )
= 5 5 − ( 5 7.29 − 5 2.916 ) = 1 − 5 4.374 = 1 − 0.8748 = 0.1252
So, 0.9) = 0.1252"> P ( X > 0.9 ) = 0.1252 .
State the final answer a) The value of b is 1.5 .
b) The probability that the depth of moisture exceeds 0.9 is 0.1252 .
Examples
Understanding probability density functions is crucial in various fields. For instance, in environmental science, it helps model rainfall patterns to predict flooding risks. Imagine using this model to determine the likelihood that rainfall exceeds a certain level, allowing for timely warnings and preventive measures to protect communities.
