Question

The number of visits to a website is known to have a Poisson distribution with a mean of 8 visits per minute. a. What is the probability distribution for \(x\), the number of visits per minute? b. What is the probability that the number of visits per minute is less than or equal to \(12 ?\) c. What is the probability that the number of visits per minute is greater than \(16 ?\) d. Within what limits would you expect the number of visits to this website to lie at least \(89 \%\) of the time?

Short answer

To summarize: a) The probability distribution for the number of visits per minute, x, is given by \(P(x) = \frac{(8^x)(e^{-8})}{x!}\). b) The probability that the number of visits per minute is less than or equal to 12 is approximately 0.996. c) The probability that the number of visits per minute is greater than 16 is approximately 0.032. d) The number of visits would lie within the range [4, 13] at least 89% of the time.

Step-by-step solution

Identify distribution and parameters

We are given that the number of visits follows a Poisson distribution with a mean of \(8\) visits per minute. Therefore, \(\lambda = 8\).

Write probability mass function

The probability mass function (PMF) of the Poisson distribution is given as \(P(x) = \frac{(\lambda^x)(e^{-\lambda})}{x!}\). Plug in \(\lambda=8\) to get the probability distribution for x, the number of visits per minute: \(P(x) = \frac{(8^x)(e^{-8})}{x!}\). #b. Probability of Number of Visits Less Than or Equal to 12#

Use Cumulative Distribution Function

To find the probability of x being less than or equal to 12, we can calculate the cumulative distribution function (CDF) up to 12: \(P(X \leq 12) = \sum_{x=0}^{12} P(x)\).

Calculate the Probability

Using the PMF we derived earlier, calculate the probabilities for \(x = 0, 1, 2, ..., 12\) and sum them up: \(P(X \leq 12) = \sum_{x=0}^{12} \frac{(8^x)(e^{-8})}{x!} \approx 0.996\). #c. Probability of Number of Visits Greater than 16#

Use Complementary Probability

To find the probability of x being greater than 16, we can use the complementary probability: \(P(X > 16) = 1 - P(X \leq 16)\).

Calculate the Probability

Using the PMF we derived earlier, calculate the probabilities for \(x = 0, 1, 2, ..., 16\) and sum them up. Then subtract the sum from 1: \(P(X > 16) = 1 - \sum_{x=0}^{16} \frac{(8^x)(e^{-8})}{x!} \approx 0.032\). #d. Limits for Number of Visits 89% of the Time#

Determine CDF Values

Find the lower and upper values of the cumulative distribution function (CDF) that include 89% of the values. The lower value is 5.5% (0.055) and the upper value is 94.5% (0.945).

Find Limits

From the calculated CDF, identify the number of visits that correspond to CDF values 0.055 and 0.945: \(P(X \leq L) = 0.055\) and \(P(X \leq H) = 0.945\), where L and H are the low and high limits, respectively. It takes a bit of trial and error to find L and H, giving: \(P(X \leq 3) = 0.042\) and \(P(X \leq 13) = 0.948\).

Report the Limits

Since we want the range where the number of visits lie at least 89% of the time, we require the smallest range that contains the 89% of values. In this case, it is \([4, 13]\) visits per minute.

Key concepts

Probability Mass Function

The probability mass function (PMF) is a fundamental concept in statistical analysis of discrete random variables. It describes the likelihood of a random variable taking on various discrete values. In the context of the Poisson distribution, the PMF is particularly relevant since we deal with the count of occurrences within a fixed interval of time or space.

For our exercise with website visits, the Poisson PMF is formally expressed as: \(P(X=x) = \frac{(\text{e}^{-\text{λ}}\text{λ}^x)}{x!}\), where \(X\) is the random variable representing the number of website visits in one minute, \(λ\) is the average number of visits (given as 8 visits per minute), \(\text{e}\) is the base of the natural logarithm, and \(x!\) is the factorial of \(x\). The factorial function, denoted as \(x!\), is the product of all positive integers less than or equal to \(x\).

This formula allows us to calculate the exact probability of having any specific number of visits, such as 0, 1, 2, and so on. For instance, if a student wanted to know the probability of exactly 5 visits in a minute, they would plug \(x = 5\) into the PMF formula.

Cumulative Distribution Function

While the PMF gives us the probability of a specific number of events, the cumulative distribution function, or CDF, serves a different purpose. It is a step higher—the CDF tells us the probability that a random variable \(X\) is less than or equal to a certain value \(x\). In essence, it's the accumulation of probabilities of the PMF up to that point.

For the Poisson distribution, we can calculate the CDF by adding up the probabilities for all values less than or equal to \(x\), as shown in our exercise for being less than or equal to 12 visits. We do that by computing \(P(X \times\timesleq 12) = \sum_{n=0}^{12} P(X=n)\), using the previously established PMF for each value of \(n\).

Understanding the CDF is critical when trying to find probabilities over a range of outcomes, such as needing to know if the number of visits is within a particular range. It's also essential when determining percentile ranks, where you might need to know the value beneath which a certain percentage of observations fall.

Complementary Probability

Sometimes it's easier to consider the opposite of what we're looking for. This is where the concept of complementary probability comes in handy. Complementary probability refers to the probability that the opposite of an event occurs—essentially, it adds up to 1 with the probability of the event itself.

For a Poisson distributed random variable, like in our example about website visits, we use complementary probability to find the likelihood of more than a set number of occurrences. For instance, to find out the probability of getting more than 16 visits in a minute, we calculate \(P(X > 16)\). Thanks to complementary probability, we can instead evaluate \(1 - P(X \times\timesleq 16)\), which is easier since we can use the CDF up to 16.

This concept is particularly useful for calculating the upper tail probabilities—where the PMF would require a lengthy addition of many probabilities, the complement rule can give the result with a single subtraction. It's an efficient tool, providing a shortcut in probability calculations, and is a key strategy in solving statistical problems where the direct approach is impractical.