1. ## Help needed please! probability distributions

1. The time (in hours) required to repair a machine is an exponentially distributed random variablewith parameter λ = 1/4.
(a) What is the probability that a repair time exceeds 4 hours.
(b) What is the conditional probability that a repair takes at least 10 hours, given that its duration exceed 9 hours.

For a) I got the answer $\displaystyle 1 - e^{-1}$

How do you do b)?
Is it $\displaystyle P(X>=10) = 1 - P(X<=9)?$

2. An electronic equipment is used to measure brain mass. If the brain mass for a randomly selected person is m grams, the machine measures and amount Y , where Y = m + X and X is normally distributed with mean 0 and variance 4. What is the probability that the measure Y does not deviate from the true value m by more than 1.3 grams?

Thank you very much!

2. Originally Posted by Ruichan
1. The time (in hours) required to repair a machine is an exponentially distributed random variablewith parameter λ = 1/4.
(a) What is the probability that a repair time exceeds 4 hours.
(b) What is the conditional probability that a repair takes at least 10 hours, given that its duration exceed 9 hours.

For a) I got the answer $\displaystyle 1 - e^{-1}$

How do you do b)?
Is it $\displaystyle P(X>=10) = 1 - P(X<=9)?$

Thank you very much!
(a) Let T be the time until finished repair. Then T is Exp(1/4) and has CDF

$\displaystyle P(T<t) = 1 - e^{-t/4}$

Probability that a repair exceeds t hours is then
$\displaystyle P(T>t) = 1 - P(T<t) = e^{-t/4}$

(b) Using the definition of conditional probability we get, for 0 < s < t

$\displaystyle P(T>t|T>s) = P(T>t,T>s)/P(T>s) = P(T>t)/P(T>s) =$

$\displaystyle = e^{-(t-s)/4}$