Convicting a Defendant: Finding the Optimal Threshold for Conviction Probability

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In summary, the value of an exponential random variable X with mean m is used to summarize evidence in a criminal investigation. The deciding judge will rule the defendant guilty if X > c for some c. To ensure a 95% chance of not convicting an innocent man, the value of c should be such that P{X <= c} = 0.95 with m = 1. With this value of c, the probability of convicting a guilty defendant is found by using P{X > c} with m = 2.
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e(ho0n3
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[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?
 
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  • #2
Yes to both of your questions.
 
  • #3
Thanks. I always have trouble interpreting probability questions.
 

1. How does a defendant get convicted?

A defendant is convicted through a legal process that involves a trial and a jury. The prosecution must present evidence and arguments to prove the defendant's guilt beyond a reasonable doubt. If the jury finds the defendant guilty, they will be convicted and sentenced accordingly.

2. What is the role of evidence in convicting a defendant?

Evidence is crucial in convicting a defendant as it is used to prove their guilt beyond a reasonable doubt. Evidence can include witness testimonies, physical evidence, and expert opinions. The type and strength of evidence presented can greatly impact the outcome of a trial.

3. Can a defendant be convicted without a trial?

In some cases, a defendant may choose to plead guilty to the charges against them. This means they accept responsibility for their actions and do not go to trial. In this scenario, the judge can enter a conviction and sentence the defendant without a trial.

4. What are some common defenses used to avoid a conviction?

Some common defenses used to avoid a conviction include lack of evidence, self-defense, and insanity. A defendant's legal team may also argue that their constitutional rights were violated during the arrest or trial, which could result in a mistrial or acquittal.

5. What happens after a defendant is convicted?

After a defendant is convicted, they will be sentenced by the judge. The sentence can range from fines and probation to imprisonment or even the death penalty, depending on the severity of the crime and the laws in the jurisdiction. The defendant also has the right to appeal their conviction to a higher court.

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