Question

1. a violent crime occurs every 10 minutes in a depressed urban area. let ( x = ) time it takes for the next violent crime to occur. find:

a. ( p(x>50) )
b. ( p(x > 30) )
c. ( p(x < 9) )
d. ( p(x < 2) )
e. ( p(5 < x < 8) )
f. ( p(16 < x < 32) )

Explanation:

Step1: Determine the rate parameter $\lambda$

Since a violent crime occurs every 10 minutes, the rate parameter $\lambda=\frac{1}{10}=0.1$. The probability density function of an exponential distribution is $f(x)=\lambda e^{-\lambda x}$ for $x\geq0$, and the cumulative - distribution function is $F(x)=P(X\leq x)=1 - e^{-\lambda x}$.

Step2: Calculate $P(X > 50)$

Using the property $P(X > x)=e^{-\lambda x}$. Substitute $\lambda = 0.1$ and $x = 50$ into the formula.
$P(X>50)=e^{-0.1\times50}=e^{-5}\approx0.0067$

Step3: Calculate $P(X > 30)$

Using the property $P(X > x)=e^{-\lambda x}$. Substitute $\lambda = 0.1$ and $x = 30$ into the formula.
$P(X>30)=e^{-0.1\times30}=e^{-3}\approx0.0498$

Step4: Calculate $P(X < 9)$

Using the cumulative - distribution function $P(X < x)=1 - e^{-\lambda x}$. Substitute $\lambda = 0.1$ and $x = 9$ into the formula.
$P(X < 9)=1 - e^{-0.1\times9}=1 - e^{-0.9}\approx1 - 0.4066 = 0.5934$

Step5: Calculate $P(X < 2)$

Using the cumulative - distribution function $P(X < x)=1 - e^{-\lambda x}$. Substitute $\lambda = 0.1$ and $x = 2$ into the formula.
$P(X < 2)=1 - e^{-0.1\times2}=1 - e^{-0.2}\approx1 - 0.8187 = 0.1813$

Step6: Calculate $P(5 < X < 8)$

Using the formula $P(a < X < b)=e^{-\lambda a}-e^{-\lambda b}$. Substitute $\lambda = 0.1$, $a = 5$, and $b = 8$ into the formula.
$P(5 < X < 8)=e^{-0.1\times5}-e^{-0.1\times8}=e^{-0.5}-e^{-0.8}\approx0.6065 - 0.4493=0.1572$

Step7: Calculate $P(16 < X < 32)$

Using the formula $P(a < X < b)=e^{-\lambda a}-e^{-\lambda b}$. Substitute $\lambda = 0.1$, $a = 16$, and $b = 32$ into the formula.
$P(16 < X < 32)=e^{-0.1\times16}-e^{-0.1\times32}=e^{-1.6}-e^{-3.2}\approx0.2019 - 0.0408 = 0.1611$

Answer:

A. $P(X > 50)\approx0.0067$
B. $P(X > 30)\approx0.0498$
C. $P(X < 9)\approx0.5934$
D. $P(X < 2)\approx0.1813$
E. $P(5 < X < 8)\approx0.1572$
F. $P(16 < X < 32)\approx0.1611$