A contractor is required by a county planning department to...

A contractor is required by a county planning department to submit one, two, three, four, or five forms (depending on the nature of the project) in applying for a building permit. Let $Y=$ 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 $k ?\left[$Hint$: \Sigma_{y}^{5}=1 p(y)=1 .\right]$ (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 $y=1, \ldots, 5$ be the pmf of $Y ?$

A contractor is required by a county planning department to submit one, two, three, four, or five forms (depending on the nature of the project) in applying for a building permit. Let $Y=$ 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 $k ?\left[$Hint$: \Sigma_{y}^{5}=1 p(y)=1 .\right]$
(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 $y=1, \ldots, 5$ be the pmf of $Y ?$

Key Concepts

Validity of a Probability Mass Function (PMF)

For a function to serve as a probability mass function, it must meet two criteria: all probabilities must be non-negative, and the sum of the probabilities over all possible outcomes must equal one. This concept is used to verify or refute any proposed function as a legitimate PMF for a discrete random variable.

Cumulative Probability

Cumulative probability refers to the probability that a random variable takes on a value less than or equal to a particular number. In the problem, calculating the probability that at most a certain number of forms are required involves summing the probabilities for all outcomes up to that number.

Normalization of Probability Distributions

This concept ensures that the total probability across all outcomes sums to one, a fundamental property of any probability distribution. In the context of the problem, the condition ? p(y) = 1 is used to find the constant of proportionality in a probability distribution that is defined up to a constant factor.

Proportional Probability Functions

A proportional probability function assigns probabilities that are proportional to a specified function of the outcome, meaning each probability is given as a constant multiplied by some function of the outcome. To determine the probabilities, the constant of proportionality must be found by applying the normalization condition.

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