Question

The loss due to a fire in a commercial building is modeled by a random variable X with density function

f(x)={k(x−2),for 2≤x<7 f(x)={k(12−x),for 7≤x<12

where k is a constant.

Determine the probability that the fire loss is between 5 and 9.

0.24

0.30

0.52

0.64

0.71

Answer 1

Answer:

Answer is 0.64

Step-by-step explanation:

Given that pdf of x is

$f(x)={k(x−2),for 2≤x<7\\ f(x)={k(12−x),for 7≤x<12$

Since total probability is 1  we can use this to find k.

$\int\limits^7_2 {k(x-2)} \, dx +\int\limits^12_7 {k(12-x)} \, dx \\=k\frac{(x-2)^2}{2} -k\frac{(12-x)^2}{2} \\=\frac{k}{2} [(7-2)^2-(2-2)^2-(12-12)^2+(7-12)^2]\\=\frac{k}{2}[25+25]=1\\k =\frac{1}{25}$

the probability that the fire loss is between 5 and 9=$P(5<x<9)=\int\limits^9_5 {f(x)} \, dx \\=\int\limits^7_5 {f(x)} \, dx+\int\limits^9_7 {f(x)} \, dx\\=(8+8)\frac{1}{25} \\=0.64$