problem 3.(25pt) the annual wind speed in a mountainous town has an exponential distribution and has a mean…

Explanation:

Step1: Find the parameter of the exponential - distribution

The mean of an exponential distribution is $\mu=\frac{1}{\lambda}$. Given $\mu = 20$, then $\lambda=\frac{1}{20}=0.05$. The probability density function of an exponential distribution is $f(x)=\lambda e^{-\lambda x}$ for $x\geq0$, and the cumulative - distribution function is $F(x)=1 - e^{-\lambda x}$.

Step2: Calculate the probability that the wind speed exceeds 50 mph in a single year

We want to find $P(X > 50)$. Using the property of the exponential distribution $P(X>x)=e^{-\lambda x}$. Substituting $\lambda = 0.05$ and $x = 50$, we get $P(X > 50)=e^{-0.05\times50}=e^{-2.5}\approx0.0821$.

Step3: Calculate the probability that the tower gets damaged in 10 years (part a)

Let $Y$ be the number of years in which the wind speed exceeds 50 mph in 10 years. $Y$ follows a binomial distribution with parameters $n = 10$ and $p=e^{-2.5}$. The probability that the tower gets damaged (i.e., $Y\geq1$) is $P(Y\geq1)=1 - P(Y = 0)$. Using the binomial probability formula $P(Y = k)=C(n,k)p^{k}(1 - p)^{n - k}$, where $C(n,k)=\frac{n!}{k!(n - k)!}$, for $k = 0$, $n = 10$, and $p=e^{-2.5}$, we have $P(Y = 0)=(1 - e^{-2.5})^{10}$. So $P(Y\geq1)=1-(1 - e^{-2.5})^{10}=1-(1 - 0.0821)^{10}=1 - 0.9179^{10}\approx1-0.434=0.566$.

Step4: Calculate the probability that the tower collapses within 10 years after it is built (part b)

The tower has already experienced winds exceeding 50 mph in two different years in the past. We want to find the probability that it experiences such a wind again within 10 years. We can use the memory - less property of the exponential distribution. The probability that the wind speed exceeds 50 mph in the next 10 years is the same as the probability in a single non - overlapping 10 - year period. Let $Z$ be the event that the wind speed exceeds 50 mph in the next 10 years. The probability that the wind speed exceeds 50 mph in a single year is $p = e^{-2.5}$. The probability that the wind speed exceeds 50 mph at least once in 10 years is $P(Z\geq1)=1 - P(Z = 0)$. Using the binomial approach with $n = 10$ and $p=e^{-2.5}$, $P(Z = 0)=(1 - e^{-2.5})^{10}$, so $P(Z\geq1)=1-(1 - e^{-2.5})^{10}\approx0.566$.

Answer:

(a) $0.566$ (b) $0.566$