Question

A contractor is required by a county planning department to submit anywhere from one to five forms (depending on the nature of the project) in applying for a building permit. Let \(y\) be the number of forms required of the next applicant. The probability that \(y\) forms are required is known to be proportional to \(y ;\) that is, \(p(y)=k y\) for \(y=1, \ldots, 5\). a. What is the value of \(k ?\) (Hint: \(\sum p(y)=1 .\) ) b. What is the probability that at most three forms are required? c. What is the probability that between two and four forms (inclusive) are required? d. Could \(p(y)=y^{2} / 50\) for \(y=1,2,3,4,5\) be the probability distribution of \(y\) ? Explain.

Short answer

a. The value of \(k\) should be calculated in step 1 as outlined above. b. The probability of requiring at most three forms should be calculated in step 2 as outlined above. c. The probability that between 2 and 4 forms are required will be calculated in step 3 as outlined above. d. The validation of the proposed alternate probability function \(p(y)=\frac{y^2}{50}\) will be done in step 4 as described above.

Step-by-step solution

Calculate the value of \(k\)

The first step requires the calculation of a constant \(k\). We know that the total probability should sum to 1, which implies \(\sum_{y=1}^{5} p(y) = 1\). Substituting \(p(y) = ky\), the equation becomes \(\sum_{y=1}^{5} ky = 1\). Solving this equation gives the value of \(k\).

Calculate the probability of requiring at most three forms

Here, the goal is to calculate the cumulative probability of \(y\) being 1, 2 or 3 forms. Formally, \(P(Y \leq 3)\). Using the function \(p(y) = ky\) derived in Step 1, add up \(p(1), p(2)\) and \(p(3)\) to find this probability.

Calculate the probability that between two and four forms are required

Here we are looking for \(P(2 \leq Y \leq 4)\). This can be determined by calculating and adding \(p(2), p(3)\) and \(p(4)\) using the \(p(y) = ky\) function determined in Step 1.

Analysis of the alternate probability function

Finally, we are required to analyze whether the function \(p(y) = \frac{y^2}{50}\) could serve as an alternate probability distribution. This will require calculating the sum of the probabilities generated by this function from \(y = 1\) to \(y = 5\), and checking if that sum is equal to 1, thereby confirming it as a valid probability distribution.

Summarizing the results

After executing steps 1 to 4, summarize the results obtained from each of these steps: the value of \(k\), the probability of y being at most 3 and the probability of y being between 2 and 4, inclusive and finally validating the alternate probability distribution function.

Key concepts

Cumulative Probability

Cumulative probability is a crucial concept in understanding the likelihood of an event occurring within a certain range. For example, if a contractor might need to submit anywhere from one to five forms, the cumulative probability helps to answer questions like, 'What is the probability that at most three forms are required?'

To compute this, one would add the individual probabilities of requiring one, two, or three forms. If the probability distribution is given by a function such as \(p(y) = ky\), where \(k\) is a constant, the cumulative probability for up to three forms, \(P(Y \leq 3)\), is the sum of \(p(1) + p(2) + p(3)\). This technique is used widely in statistical analysis to predict outcomes within a specified range, and is a foundation for creating cumulative distribution functions which graphically represent these probabilities over the entire range of possible outcomes.

Probability Function

A probability function, or probability mass function (PMF) in the context of a discrete random variable, is a function that gives the probability that a discrete random variable is exactly equal to some value. The function provided in our example, \(p(y) = ky\), where \(y\) is the number of forms required by the contractor, is a probability function for the discrete random variable \(y\).

To determine the constant \(k\), one must use the fact that the sum of all probabilities must equal 1. So, setting up an equation where \(k\) is multiplied by each value of \(y\) and summing it all to equal 1 provides one with the necessary value of \(k\). With this function, one can calculate individual probabilities and, as explained earlier, cumulative probabilities as well.

Discrete Random Variable

A discrete random variable is one that has a countable number of possible values. In the scenario of our building permit applicant, the variable \(y\), representing the number of forms a contractor must submit, is a discrete random variable because \(y\) can only take on the values {1, 2, 3, 4, 5}.

When dealing with discrete random variables, the probability function is especially handy as it precisely assigns probabilities to these countable outcomes. It's imperative for statistical analysis to categorize variables correctly as discrete or continuous in order to apply appropriate methods to predict or understand behavior and outcomes in a given situation.

Statistical Analysis

Statistical analysis encompasses a broad set of methods for exploring, describing, and drawing inferences from data. When analyzing a probability distribution like \(p(y) = ky\) for our discrete random variable \(y\), statisticians engage in a qualitative and quantitative examination to derive meaningful insights, such as expected values, variance, and other characteristics of the distribution.

Moreover, when questioning if an alternate probability function, such as \(p(y) = \frac{y^2}{50}\), could be appropriate, statistical analysis involves checking the validity of the distribution by ensuring the total probability sums to 1 across all possible values of \(y\). Through statistical analysis, one can assess the likelihood of various outcomes, predict future events, and make informed decisions based on data.