Q.5.19 - A First Course in Probability

Question

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

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

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

Step-by-Step Solution

Answer

1. If 95 percent certain that an innocent man will not be convicted then the value of c will be $2.996$.
2. The probability that a guilty defendant will be convicted is $0.2236$.

Step 1: Given information (Part a)

Variable $X$ whose mean $\mu$ depends on whether the defendant is guilty

Innocent $\mu = 1$

Guilty $\mu = 2$

Step 2: Solution (Part a)

The probability density function will be,

$f\left(x\right)=\left\{\begin{array}{c}{e}^{-x},x\ge 0\\ 0\text{ otherwise }\end{array}\right.$

The judge wants to be 95 percent certain. In other words, the judge wants to rule the defendant guilty for values of $X>c$. Hence, one needs to find the value of c such that:

$P\left\{X_1>c\right\}=0.05$

$\Rightarrow {\int }_c^{\mathrm{\infty }} e^{-x}dx=0.05$

$\Rightarrow {\left(e^{-x}\right)}_c^{\mathrm{\infty }}=0.05$

$\Rightarrow e^{-c}=0.05$

$\Rightarrow e^{-c}=\frac{1}{20}$

By applying the logarithm on both sides, we get

$c=\log \left(20\right)$

$=2.996$

Step 3: Final answer (Part a)

If 95 percent certain that an innocent man will not be convicted then the value of c will be $2.996$

Step 4: Given information (Part b)

Variable $X$ whose mean $\mu$ depends on whether the defendant is guilty

Innocent $\mu =1$

Guilty $\mu =2$

Step 5: Solution (Part b)

The probability density function will be,

$f\left(x\right)=\left\{\begin{array}{l}\frac{1}{2}{e}^{-\frac{1}{2}x},x\ge 0\\ 0\text{ otherwise }\end{array}\right.$

The deciding judge will rule the defendant guilty if $X_2>c$. Hence, the probability that a guilty defendant will be convicted is:

$P\left\{X_2>c\right\}=e^{-\frac{c}{2}}$

$\Rightarrow {\int }_c^{\mathrm{\infty }} \frac{1}{2}{e}^{-\frac{1}{2}x}dx=0.05$

$\Rightarrow {\left(e^{-\frac{1}{2}x}\right)}_c^{\mathrm{\infty }}=0.05$

$\Rightarrow e^{-\frac{c}{2}}=\frac{1}{20}$

But from part (a), it was shown that:

$e^{-c}=\frac{1}{20}$

$e^{-\frac{1}{2}c}=\frac{1}{\sqrt{20}}$

The probability that a guilty defendant will be convicted guilty is,

$P\left\{X_2>c\right\}=\frac{1}{\sqrt{20}}$

$=0.2236$

Step 6: Final answer (Part b)

The probability that a guilty defendant will be convicted is $0.2236$.