Question

3. From 2000-2019 there were a total of 3071 earthquakes worldwide with a magnitude of 6 or greater, or an average of about 0.42 such earthquakes per day.* Assume that moving forward the total number of such earthquakes to occur over any time period follows a Poisson distribution with an average of 0.42 earthquakes per day. For the remainder of this question, "earthquake" will mean an earthquake with a magnitude of 6 or greater. Define a new random variable as necessary in each part of this question. (a) What is the probability that there are no earthquakes during a single day? (b) What is the probability that there are at least three earthquakes during a single week? FUN FACT: When the number of events over any time interval follow a Poisson distribution, the time between any two events follows an exponential distribution with a mean equal to the reciprocal of the mean for the Poisson distribution. Therefore the time between two earthquakes follows an exponential distribution with an average of about 2.38 days. Answer the following three questions using the exponential distribution. (c) What is the probability that the time between two earthquakes will be less than three days? (d) If an earthquake just occurred, what is the probability that the time until the next earthquake will be more than 12 hours but less than 24 hours? (e) What is the median time between two earthquakes?

Answer 1

Answer:-

Given That:-

From 2000-2019 there were a total of 3071 earthquakes worldwide with a magnitude of 6 or greater, or an average of about 0.42 such earthquakes per day.* Assume that moving forward the total number of such earthquakes to occur over any time period follows a Poisson distribution with an average of 0.42 earthquakes per day.

Given,

The number of earthquakes over a period of time follows poisson distribution with average = 0.42 earth quakes per day.

(a) What is the probability that there are no earthquakes during a single day?

P(X = 0) in a single day:

Formula:-

$P(X=x)=\frac{\lambda^x*e^{-\lambda}}{x!}$

Here

$\lambda$ = 0.42 per day

x = 0

$P(X=0)=\frac{(0.42)^0*e^{-0.42}}{0!}$

= 0.6571

Therefore,

Probability that there are no earth quakes during a single day = 0.6571 days

(b) What is the probability that there are at least three earthquakes during a single week?

in a week:

P(X > 3

Now

$\lambda$ = average number of earth quakes in a week

= 0.42* 7

$\lambda$ = 2.94

x takes 3, 4, 5, 6, ----

OX

Therefore,

$P(X \geq 3)=\sum_{x=3}^{\infty}\frac{\lambda^x*e^{-\lambda}}{x!}$

$P(X \geq 3)=\sum_{x=3}^{\infty}\frac{(2.94)^x*e^{-2.94}}{x!}$ [Because Using poisson distribution online calculator]

= 0.5587

Therefore,

Probability that there are atleast 3

Earth quakes in a single week = 0.5587

We know that if number of earth quakes occur over a period of time follows poisson distribution with average  $\lambda$ then the time between any two events follows an exponential distribution with average 1/$\lambda$

= 1/0.42

= 2.38 days

=  $\mu$

(c) What is the probability that the time between two earthquakes will be less than three days?

P(X < 3 days):-

Formula:-

$P(X <x) = 1 - e^{\frac{-1}{\mu}*x}$

Here x takes 0, 1, 2, days

$\mu$ = 2.38

$P(X <3) = 1 - e^{\frac{-1}{2.38}*3}$

= 0.7165 [Because Using ordinary calculator]

Therefore

Probability that time between two successive earth quakes is less than 3 days = 0.7165

(d) If an earthquake just occurred, what is the probability that the time until the next earthquake will be more than 12 hours but less than 24 hours?

P( 12 hr < X < 24 hr) i.e, P(1/2 day < X < 1 day)

P(1/2 < X < 1) = P(X < 1) - P(X  $\leq$ 1/2)

So,

$P(X<1)=1-e^{\frac{1}{\mu}x}$

$P(X<1)=1-e^{\frac{1}{2.38}1}$

$P(X\leq 1/2) =1-e^{-1/2.38*1/2}$

$P(X\leq 1/2) =1-e^{-1/4.76}$

$P(1/2<X<1)=P(X<1)-P(X\leq 1/2)$

$P(1/2<X<1)=(1-e^{-1/238})-(1-e^{-1/476})$

$P(1/2<X<1)=e^{-1/476}-e^{-1/238}$

= 0.8105 - 0.6569

= 0.1536

Therefore,

Probability that the time between two successive earth quakes is more than 12 hours but less than 24 hrs = 0.1536

(e) What is the median time between two earthquakes?

Median time between two earth quakes:

Median time = $\mu$ * ln 2

= 2.38 * ln 2

= 1.65 days

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