Question

The lifetime of a machine part has a continuous distribution on the interval​ (0, 50​) with probability density function​ f, where​ f(x) is proportional to left parenthesis 10 plus x right parenthesis Superscript negative 2. Calculate the probability that the lifetime of the machine part is less than 13.

Answer 1

Probability that the lifetime of the machine part is less than 13 = 0.6782

Explanation:

given that
`f(x)=(10+x)^(-2)`

Normalizing the function we get

`\int_(0 )^(50)cf(x)dx=1`

`\int_(0 )^(50)c(10+x)^(-2)dx=1`

`\therefore a=(1)/(\int_(0 )^(50)(10+x)^(-2)dx)`

`\therefore a=12`

`P(x< 13)=\int_(0)^(13)12(10+x)^(-2)dx\\\\P(X< 13)=0.6782`