In the paper "The Distribution of Wars in Time" (Journal of...

In the paper "The Distribution of Wars in Time" (Journal of the Royal Statistical Society, Vol. 107, No. 3 / 4 pp. 242-250), L. F. Richardson analyzed the distribution of wars in time. From the data, we determined that the number of wars that begin during a given calendar year has roughly a Poisson distribution with parameter $\lambda=0.7 .$ If a calendar year is selected at random, find the probability that the number, $X,$ of wars that begin during that calendar year will be a. zero. b. at most two. c. between one and three, inclusive. d. Find and interpret the mean of the random variable $X.$ e. Determine the standard deviation of $X.$

In the paper "The Distribution of Wars in Time" (Journal of the Royal Statistical Society, Vol. 107, No. 3 / 4 pp. 242-250), L. F. Richardson analyzed the distribution of wars in time. From the data, we determined that the number of wars that begin during a given calendar year has roughly a Poisson distribution with parameter $\lambda=0.7 .$ If a calendar year is selected at random, find the probability that the number, $X,$ of wars that begin during that calendar year will be
a. zero.
b. at most two.
c. between one and three, inclusive.
d. Find and interpret the mean of the random variable $X.$
e. Determine the standard deviation of $X.$

Key Concepts

Poisson Distribution

The Poisson distribution is a discrete probability distribution used to model the number of times an event occurs in a fixed interval of time or space when these events happen at a constant mean rate and independently of the time since the last event. It is useful in various fields for predicting the probability of a given number of events happening over a period of time.

Probability Mass Function (PMF)

The probability mass function for a Poisson distribution provides the probability of observing exactly k events in a given interval. It is defined as P(X=k) = (e^(-?) * ?^k) / k!, where ? is the average rate of occurrence. This formula allows for the computation of exact probabilities for specific counts of events.

Cumulative Probability

Cumulative probability in the context of a Poisson distribution is computed by summing the probabilities obtained from the PMF for all values up to a selected threshold. This is important for finding the probability that the random variable takes on a value within a range (or at most a certain value), rather than exactly a specific value.

Mean and Variance

In a Poisson distribution, a key property is that both the mean and the variance are equal to the parameter ?. The mean represents the expected number of events occurring in the interval, while the variance measures how spread out the number of events can be around the mean. This equality simplifies analysis and interpretation of the distribution.

Standard Deviation

The standard deviation of a Poisson distribution is the square root of its variance, which is ??. This measure provides insight into the typical deviation from the mean number of events and is crucial for understanding the variability inherent in the process being modeled.

Transcript

00:01 We're modeling the number of wars that begin in any year, and we're told that this can be modeled as a poisson distribution with our parameter lambda equals 0 .7.

00:12 And we're asked a few probabilities for the number of wars beginning in a year.

00:17 Probability that no wars start, probability that less than or equal to two wars start, probability that between one and three wars start, and also what the mean and standard deviation for this probability of war beginning in a given year is.

00:37 So for this problem, what we need to know, we need to know the equation for probability in a poisson distribution, and then we'll use that to calculate the probabilities in parts a, b, and c, and that's pretty straightforward.

00:55 And we also need to know the definition of mean and standard deviation for a poisson distribution for part d and e.

01:04 And that's very, very simple.

01:08 Okay, so for our poisson distribution, the probability that our random variable is equal to x can be described as e to the minus lambda.

01:25 Lambda is this parameter at times lambda to the x over x factorial.

01:35 And here we have, we're given lambda equals 0 .7 for this problem.

01:47 All right, part a.

01:48 We want the probability of x being equal to zero.

01:55 And so that's going to be equal to e to the minus 0 .7 times 0 .7 to the 0 over 0 factorial.

02:09 And so we want to remember that 0 factorial is defined as 1, since it's the number of ways to arrange things.

02:18 So it's the same situation as 1 factorial.

02:21 There's one of them, and there's only one way you can arrange it.

02:24 And so that will work out to be e to the minus 0 .7 times 1 over 1.

02:33 And that works out to be a little bit under one half.

02:39 So the probability that there will not be a war starting in any given year is about 50%.

02:49 And part b, we want the probability that the probability that, that less than or equal to two words start in a given year...

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