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Chapter 5: Problem 1

Contracts for two construction jobs are randomly assigned to one or more of three firms, A. B. and C. Let Y 1 denote the number of contracts assigned to firm A and Y 2 the number of contracts assigned to firm B. Recall that each firm can receive \(0.1 .\) or 2 contracts. a. Find the joint probability function for \(Y_{1}\) and \(Y_{2}\). b. Find \(F(1,0)\).

Short Answer

Expert verified
F(1,0) is the probability that \(Y_1 \leq 1\) and \(Y_2 \leq 0\). Calculate from joint probabilities.

Step by step solution

01

Define the Sample Space

Start by defining the possible number of contracts that can be allocated to each firm, given we have two contracts in total. Each firm can receive 0, 1, or 2 contracts. Let鈥檚 create the sample space for all possible assignments of these contracts among the three firms. Each outcome must sum to 2 (total number of contracts):\[(Y_1, Y_2, Y_3) \text{ such that } Y_1 + Y_2 + Y_3 = 2\].
02

Determine the Joint Probability Distribution

Calculate the probability for each possible pair (\(Y_1, Y_2\)) based on the sample space and uniform distribution assumption (if there's no preference given). The possible pairs are (0,0), (0,1), (0,2), (1,0), (1,1), (1,2), (2,0), (2,1), (2,2). Assign probabilities assuming equal likelihood, for example, if using a three-sided fair die model to randomly assign contracts between the firms.
03

Construct the Joint Probability Table

Create a table with rows as values of \(Y_1\) (number of contracts assigned to firm A) and columns as values of \(Y_2\) (number of contracts assigned to firm B). Enter the probabilities calculated in the previous step for each pair into this table.
04

Calculate F(1,0)

The value \(F(1,0)\) refers to the cumulative distribution function value at \(Y_1 = 1\) and \(Y_2 = 0\). This is the sum of probabilities for all the pairs \((Y_1, Y_2)\) where \(Y_1 \leq 1\) and \(Y_2 \leq 0\). Based on our probability distribution, look up these values in the joint probability table and sum them up.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cumulative Distribution Function
A cumulative distribution function, often abbreviated as CDF, is a fundamental concept in probability theory. It describes the probability that a random variable takes on a value less than or equal to a certain level. In simpler terms, it accumulates probabilities from a distribution.

For a joint distribution, the CDF, denoted by \( F(x, y) \) for random variables \( Y_1 \) and \( Y_2 \), is calculated by adding up the probabilities of all possible outcomes where \( Y_1 \leq x \) and \( Y_2 \leq y \). This allows us to understand the probability of different combinations of contract assignments among firms.
  • The CDF gives us an overarching view of how the overall probabilities accumulate over the variables.
  • Mathematically, it is summed using the joint probability table created from identified outcomes like (0, 0), (0, 1), and so on.
Understanding the cumulative nature of this function helps us gauge the likelihood of various scenarios that align with a given distribution.
Probability Theory
Probability theory is the branch of mathematics dealing with the analysis of random phenomena. It is crucial for understanding how contract allocations among firms can occur.

In the context of joint probability distributions, several important principles come into play:
  • Random Experiment: Assigning contracts randomly among three firms is an example of a random experiment, where each possible outcome is inherently uncertain.
  • Sample Space: This is the set of all possible outcomes, such as \((Y_1, Y_2, Y_3)\) combinations that add up to 2.
  • Event: Any subset of the sample space is an event, like having firm A receive 1 contract while firm B receives none.
Probability theory allows us to assess these outcomes and calculate how likely it is for each to happen, guiding the creation of joint probability tables and understanding distribution decisions.
Random Variables
In probability and statistics, random variables are used to quantify outcomes of random phenomena. Here, they help in determining the number of contracts received by each firm.

Each assignment of contracts corresponds to specific values these random variables can assume. In the problem,
  • \( Y_1 \): Represents the number of contracts assigned to firm A.
  • \( Y_2 \): Represents the number of contracts assigned to firm B.
Random variables serve as the bedrock for calculating joint probabilities and distributions:
  • They help kind out how different numbers of contracts among firms could be manifested.
  • These variables are essential in constructing the joint probability distribution, as each possible combination of \( Y_1 \) and \( Y_2 \) reflects a state of nature for firms A, B, and C.
Understanding random variables' role highlights how outcomes tie directly to the broader concepts of probability distributions.

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Most popular questions from this chapter

When commercial aircraft are inspected, wing cracks are reported as nonexistent, detectable, or critical. The history of a particular fleet indicates that \(70 \%\) of the planes inspected have no wing cracks, \(25 \%\) have detectable wing cracks, and \(5 \%\) have critical wing cracks. Five planes are randomly selected. Find the probability that a. one has a critical crack, two have detectable cracks, and two have no cracks. b. at least one plane has critical cracks.

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Suppose that \(Y_{1}\) and \(Y_{2}\) have correlation coefficient \(\rho=.2 .\) What is is the value of the correlation coefficient between a. \(1+2 Y_{1}\) and \(3+4 Y_{2} ?\) b. \(1+2 Y_{1}\) and \(3-4 Y_{2} ?\) c. \(1-2 Y_{1}\) and \(3-4 Y_{2} ?\)

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Suppose that a company has determined that the the number of jobs per week, \(N\), varies from week to week and has a Poisson distribution with mean \(\lambda\). The number of hours to complete each job, \(Y_{i},\) is gamma distributed with parameters \(\alpha\) and \(\beta\). The total time to complete all jobs in a week is \(T=\sum_{i=1}^{N} Y_{i} .\) Note that \(T\) is the sum of a random number of random variables. What is a. \(E(T | N=n) ?\) b. \(E(T)\), the expected total time to complete all jobs?
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