Write down the probability function for your answer in a.) mass
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
![### Analyzing Pothole Occurrence on a Major Highway
Potholes on a major highway in the city of Chicago occur at the rate of 3.4 per mile. Below are some questions and explanations to understand the statistical distribution and characteristics of potholes over a randomly selected 3-mile stretch of highway.
#### a) Modeling Pothole Occurrence
**Question**: What random variable models the number of potholes over a randomly selected 3-mile stretch of highway?
**Answer**: The number of potholes over a randomly selected 3-mile stretch of highway can be modeled using a Poisson random variable. This is because the Poisson distribution is typically used for counting the number of events that occur in a fixed interval of time or space where the events occur independently and at a constant rate.
#### b) Probability Mass Function
**Question**: Write down the probability mass function for your answer in a).
**Answer**: The probability mass function (PMF) of a Poisson random variable \( X \) with parameter \( \lambda \) (where \( \lambda \) is the expected number of events) is given by:
\[ P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} \]
Since potholes occur at a rate of 3.4 per mile, for a 3-mile stretch, the rate \( \lambda \) is:
\[ \lambda = 3.4 \times 3 = 10.2 \]
So, the PMF for the number of potholes \( X \) over a 3-mile stretch is:
\[ P(X = k) = \frac{10.2^k e^{-10.2}}{k!} \]
#### c) Probability of Fewer than 3 Potholes
**Question**: What is the probability of fewer than 3 potholes over a 3-mile stretch of randomly selected highway?
**Answer**: To find the probability of fewer than 3 potholes (i.e., \( P(X < 3) \)), we sum the probabilities for \( X = 0, 1, \) and \( 2 \):
\[ P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2) \]
Using the PMF from above:
1. For \( X = 0 \):
\[ P(X =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F45fdd75f-924d-4ac4-9d15-03fc1f26bc02%2F79eb87db-82af-4f15-837f-2caa7e601c31%2Fwtzq9w_processed.png&w=3840&q=75)
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