Answer to Question #120453 in Statistics and Probability for aryan

Question

The survival time, in weeks, of a component X follows an exponential distribution with
parameter β = 5. (i)What is the probability that the survival time will exceed 10 weeks? (ii) What is the conditional probability that the component will survive at least 8 weeks given that it is working at the end of 3rd week? (iii) What is probability that the component will survive between 6 and 12 weeks?

Answer 1

Given $X\sim Exp(\beta), \beta=5$

$$f(x;\beta)= \begin{cases} {1\over \beta}e^{-{x\over \beta}} &\ x\geq 0 \\ 0 &\text{otherwise} \end{cases}$$

$$P(X>t)=e^{-{t\over \beta}}$$

(i)What is the probability that the survival time will exceed 10 weeks?

$$P(X>10)=e^{-{10\over 5}}=e^{-2}\approx0.135335$$

(ii) What is the conditional probability that the component will survive at least 8 weeks given that it is working at the end of 3rd week?

$$P(X\geq8|X\geq3)=P(X\geq8-3)=P(X\geq 5)=$$

$$=e^{-{5\over 5}}=e^{-1}\approx0.367879$$

(iii) What is probability that the component will survive between 6 and 12 weeks?

$$P(6<X<12)=1-P(X\geq 12)-(1-P(X\geq6)=$$

$$=P(X\geq6)-P(X\geq12)=e^{-{6\over 5}}-e^{-{12\over 5}}\approx0.210476$$