5. Binomial Distribution & Discrete Random Variables

Binomial Distribution

5. Binomial Distribution & Discrete Random Variables

Binomial Distribution

2:51 minutes

Problem 3

Textbook Question

For the distribution described in Exercise 1, find the probability of exactly 2 arrivals in one thousandth of a minute.

Verified step by step guidance

Identify the type of probability distribution: Since the problem involves counting the number of arrivals in a fixed interval of time, this is a Poisson distribution. The Poisson distribution is used to model the number of events occurring in a fixed interval of time or space when the events occur independently and at a constant average rate.

Write down the Poisson probability mass function (PMF): The PMF is given by P(X = k) = (λ^k * e^(-λ)) / k!, where λ is the average rate of arrivals in the given interval, k is the number of arrivals, and e is the base of the natural logarithm (approximately 2.718).

Determine the value of λ: λ represents the expected number of arrivals in the interval. If the average rate of arrivals per minute is given in Exercise 1, multiply that rate by the length of the interval (one thousandth of a minute) to calculate λ.

Substitute k = 2 and the calculated λ into the PMF formula: Replace k with 2 and λ with the value you calculated in the previous step. The formula becomes P(X = 2) = (λ^2 * e^(-λ)) / 2!.

Simplify the expression: Compute the factorial of 2 (2! = 2), raise λ to the power of 2, and multiply by e^(-λ). Divide the result by 2 to find the probability of exactly 2 arrivals in one thousandth of a minute.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Poisson Distribution

The Poisson distribution is a probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, given a known average rate of occurrence. It is particularly useful for modeling rare events, such as arrivals or occurrences, in a defined period. The formula for the Poisson probability mass function is P(X=k) = (λ^k * e^(-λ)) / k!, where λ is the average rate of occurrence, k is the number of events, and e is Euler's number.

Rate of Arrival (λ)

In the context of the Poisson distribution, λ (lambda) represents the average rate of arrivals or occurrences in a specified time frame. It is a crucial parameter that determines the shape of the distribution. For example, if we expect an average of 5 arrivals in one minute, λ would be 5. When calculating probabilities for shorter intervals, such as one thousandth of a minute, λ must be adjusted accordingly to reflect the expected number of arrivals in that brief period.

Probability Mass Function (PMF)

The Probability Mass Function (PMF) is a function that gives the probability that a discrete random variable is exactly equal to some value. In the case of the Poisson distribution, the PMF allows us to calculate the probability of observing a specific number of events, such as exactly 2 arrivals. By substituting the appropriate values of k and λ into the PMF formula, we can determine the likelihood of that event occurring within the defined time interval.

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b. Find the probability of exactly 40 first lines for Democrats.

Textbook Question

c. Find the probability of 40 or more first lines for Democrats.

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For groups of five drivers, find the mean and standard deviation for the numbers of drivers who say that they text while driving.

Textbook Question

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Textbook Question

In Exercises 5–8, assume that the Poisson distribution applies; assume that the mean number of Atlantic hurricanes in the United States is 5.5 per year, as in Example 1; and proceed to find the indicated probability.

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a. Find the probability that in a year, there will be 7 hurricanes.

Textbook Question

Hurricanes

b. In a 118-year period, how many years are expected to have 7 hurricanes?

Textbook Question

Hurricanes

a. Find the probability that in a year, there will be no hurricanes.

Textbook Question

Hurricanes

b. In a 118-year period, how many years are expected to have no hurricanes?

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