Question

The probability density function of the time to failure of an electronic component in a copier (in hours) is f(x)= e^-x/100 /1000. Determine the probability that



a. A component lasts more than 3000 hours before failure.


b. A component fails in the interval from 1000 to 2000 hours.


c. A component fails before 1000 hours


d. Determine the number of hours at which 10% of all components have failed.


e. Determine the cumulative distribution function for the distribution. Use the cumulative distribution function to determine the probability that a component lasts more than 3000 hours before failure.

Answer 1

Answer:

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Step-by-step explanation:

The fundamentals

A continuous random variable can take infinite values in the range associated function of that variable. Consider $f\left( x \right)f(x)$ is a function of a continuous random variable within the range $\left[ {a,b} \right][a,b]$ , then the total probability in the range of the function is defined as:

$\int\limits_a^b {f\left( x \right)dx} = 1 a∫b​ f(x)dx=1$

The probability of the function $f\left( x \right)f(x)$ is always greater than 0. The cumulative distribution function is defined as:

$F\left( x \right) = P\left( {X \le x} \right)F(x)=P(X≤x)$

The cumulative distribution function for the random variable X has the property,

$0 \le F\left( x \right) \le 10≤F(x)≤1$

The probability density function for the random variable X has the properties,

$\\\begin{array}{c}\\{\rm{ }}f\left( x \right) \ge 0\\\\\int\limits_{ - \infty }^\infty {f\left( x \right)dx} = 1\\\\P\left( E \right) = \int\limits_E {f\left( x \right)dx} \\\end{array} f(x)≥0$

Kindly check the attached image below to see the full explanation to the question above.