Poisson: Deaths Currently, an average of 7 residents of the vi...

Question

Poisson: Deaths Currently, an average of 7 residents of the village of Westport (population 760) die each year (based on data from the U.S. National Center for Health Statistics).

c. Find the probability that on a given day, there is more than one death.

Accepted Answer

Welcome back, everyone. In a small city, an average of 15 cars are reported stolen each month. The probability distribution follows the Poisson model. What is the probability that on a given day, more than one car is stolen? A says 0.0902, B 0.9002. C 0.2900. And D 0.0092. Let's recall the formula, the probability of X being equal to lowercase x in the quason probability distribution. is equal to lambda raises the power of X, multiplied by E raises the power of negative lambda divided by x factorial. We want to identify the probability that on a given day more than one car is stolen, so the probability of X being greater than 1, and we can use the method of complements. Basically, the probability of X being greater than 1 is equal to 1 minus the probability of an opposite event. So essentially, We can say that this is equal to 1 minus the probability of X being less than or equal to 1, and there are only two values which we must use. Those would be the probability of X being exactly 1 and the probability of X being 0. Right? So we're subtracting the probability of X being equal to 1 and the probability of X being equal to 0 from 1. We're using exclusive or, right? So now what we want to do is simply express P X equals 1 and P X equals 0 using the formula. But to do that, we need to identify lambda and X. So, for the X values, we already know that they are 1 and 0 respectively, and lambda is our expected value or our mean value. But since we're looking at the units of days, we're going to convert the units for lambda. So we're given 15 cars, and we have to divide by 30 days in 1 month to get the average number of cars stolen per day, right? That would be 0.5. Cars per day. And now that we have our lambda, we can express the probability of X being equal to 1. Which is lambda, that's 0.5, raise to the power of 1, multiplied by E raise to the power of -0.5 divided by one factorial. And also the probability of X being 0 is equal to 0.5, raise the power of 0. Multiplied by E raised the power of -0.5 divided by zero factorial. So now let's substitute those into the first formula. The probability of X being greater than 1. is equal to 1 minus. First of all, we are subtracting 0.5 multiplied by E raises the power of negative 0.5. We're dividing by one factorial, so there is no necessity to include. That denominator. And now we are also subtracting. 0.5 raises the power of 0, that's 1.0 factorial is 1. So we're basically subtracting another E raises the power of negative 0.5. Overall, that's 1 minus. 1.5 he raised the power of -0.5. And that is equal to 0.0902. Which corresponds to the answer choice A. Thank you for watching.

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Poisson: Deaths Currently, an average of 7 residents of the village of Westport (population 760) die each year (based on data from the U.S. National Center for Health Statistics).

c. Find the probability that on a given day, there is more than one death.

Key Concepts

Poisson Distribution

Rate Parameter (λ)

Cumulative Probability

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Poisson: Deaths Currently, an average of 7 residents of the village of Westport (population 760) die each year (based on data from the U.S. National Center for Health Statistics).c. Find the probability that on a given day, there is more than one death.