Convicting a Defendant: Finding the Optimal Threshold for Conviction Probability

[e(ho0n3]

[SOLVED] Convicting a Defendant

Homework Statement
Evidence concerning the guilt of a defendant in a criminal investigation can be summarized by the value of an exponential random variable X whose mean m depends on whether the defendant is guilty: If innocent, m = 1; if guilty, m = 2. The deciding judge will rule the defendant guilty if X > c for some c.

(a) If the judge wants to be 95 % certain that an innocent man will not be convicted, what should the value of c be?

(b) Using the value of c in (a), what is the probability that a guilty defendant is convicted?

The attempt at a solution
Concerning (a), is 95 % the probability that an innocent man will not be convicted? In other words, I have to find c such that P{X <= c} = 0.95 right and use m = 1 right?

Concerning (b), all I need to do is find P{X > c} with the c value of (a) and m = 2 right?

 

[HallsofIvy]

Yes to both of your questions.

 

[e(ho0n3]

Thanks. I always have trouble interpreting probability questions.