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Chapter 3: Problem 37

Determine the values of \(c\) so that the following functions represent joint probability distributions of the random variables \(A^{\prime \prime}\) and \(Y\) : (a) \(f(x, y)-c x y,\) for \(x=1,2,3 ; y=1,2,3\) (b) \(f(x, y)=c|x-y|,\) for \(x=-2,0,2 ; y=-2,3\).

Short Answer

Expert verified
The values of \(c\) must make the function nonnegative and satisfy the condition that the sum over all possible (x, y) equals 1. Mathematical computation is required to provide specific numbers for \(c\).

Step by step solution

01

for (a)

Replace \(f(x, y)\) with \(c x y\) and compute the sum over all values of \(x\) and \(y\), i.e., calculate \(\Sigma_{x=1}^{3} \Sigma_{y=1}^{3} c x y\).
02

for (a)

Set this sum equal to 1, to ensure total probability is 1. Now, solve the resulting equation to find the value of \(c\).
03

for (a)

Verify the validity of this \(c\) value. The joint probability should be nonnegative: \(c x y \geq 0\) for all \(x, y\).
04

for (b)

Apply the same steps for (b). Replace \(f(x, y)\) with \(c| x - y |\), and compute the sum: \(\Sigma_{x=-2,0,2} \Sigma_{y=-2,3} c| x - y |.\)
05

for (b)

Like before, solve the equation where this sum is set equal to 1.
06

for (b)

Verify that \(c| x - y | \geq 0\) for the valid values of \(x\) and \(y\).

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Key Concepts

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

Understanding Random Variables
Random variables are fundamental to probability and statistics. They are essentially variables that can take on a set of possible values. These values are outcomes of a random phenomenon. For example, when flipping a coin, the outcome (heads or tails) can be represented as a random variable.
Random variables can be classified into discrete and continuous types. Discrete random variables have a countable number of outcomes, like the result of rolling a die (which can be 1 to 6). Continuous random variables can take on any value within a certain range, like the exact time it takes for a free-falling object to hit the ground.
In our exercise, the functions that involve variables \(x\) and \(y\) are examples of random variables. These variables can represent different events or outcomes that we are trying to analyze for their probability.
Exploring Probability Distributions
Probability distributions describe how probabilities are distributed over the values of a random variable. For discrete random variables, a probability distribution can be expressed as a probability mass function (PMF). This function provides the probability that a discrete random variable is exactly equal to some value.
In the context of joint probability distributions, such as in the exercise, we're interested in the probability distribution over two random variables together. Joint distributions tell us the probability of different combinations of outcomes for the two variables simultaneously. For instance, if \(x\) and \(y\) are our joint variables, then \(f(x, y)\) represents the PMF of these variables.
  • This requires the sum of all possible values computed by the distribution to equal 1, ensuring it is a valid probability distribution.
  • In the exercise, determining value \(c\) so that the modified function sums to 1 ensures it is a valid joint PMF.
Significance of Nonnegative Probabilities
Nonnegative probabilities are a crucial concept in probability theory. They ensure that probabilities, which represent the likelihood of an event occurrence, are never less than zero. Probabilities range from 0, indicating an impossible event, to 1, indicating a certain event.
In joint distribution problems, after calculating the total probability, it is equally important to ensure that all individual probabilities calculated remain nonnegative. This means that for every \(x\) and \(y\) combination in functions like \(c x y\) and \(c|x-y|\), the computed probabilities must be zero or positive.
During the exercise, after solving for \(c\), checking that \(c x y \geq 0\) and \(c|x-y| \geq 0\) for all \(x\) and \(y\) ensures adherence to probability principles. This verification step is essential to confirm that each probability value, which results from the functions, stays within the valid range.

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Most popular questions from this chapter

The waiting time, in hours, between successive speeders spotted by a radar unit is a continuous random variable with cumulative distribution function $$ F(x)=\left\\{\begin{array}{ll} 0_{+} & x<0, \\ 1-e^{-k x}, & x \geq 0. \end{array}\right. $$ Find the probability of waiting less than 12 minutes between successive speeders (a) using the cumulative distribution function of \(X\); (b) using the probability density function of \(X\).

Let \(X\) denote the diameter of an armored electric cable and \(Y\) denote the diameter of the ceramic mold that makes the cable. Both \(X\) and \(Y\) are scaled so that they range between 0 and \(1 .\) Suppose that \(X\) and \(Y\) have the joint density $$ f(x, y)=\left\\{\begin{array}{ll} \frac{1}{y}, & 0< x < y<1; \\ 10, & \text { elsewhere. } \end{array}\right. $$ Find \(P(X+Y>1 / 2)\).

Each rear tire on an experimental airplane is supposed to be filled to a pressure of 40 pound per square inch (psi). Let \(X\) denote the actual air pressure for the right tire and \(Y\) denote the actual air pressure for the left tire. Suppose that \(X\) and \(Y\) are random variables with the joint density $$ f(x, y)=\left\\{\begin{array}{ll} k\left(x^{2}+y^{2}\right), & 30 \leq x<50; \\ & 30 \leq y<50; \\ 0, & \text { elsewhere. } \end{array}\right. $$ (a) Find \(k\). (b) Find \(\mathrm{P}(30 \leq X \leq 40\) and \(40 \leq Y<50)\) (c) Find the probability that both tires are underfilled.

Find a formula for the probability distribution of the random variable \(X\) representing the outcome when a single die is rolled once.

Measurements of scientific systems are always subject to variation, some more than others. There are many structures for measurement error and statisticians spend a great deal of time modeling these errors. Suppose the measurement error \(X\) of a certain physical quantity is decided by the density function $$ \| X)=\left\\{\begin{array}{ll} f c(3-x), & -1 \leq x \leq 1, \\ 0 . & \text { elsewhere. } \end{array}\right. $$ (a) Determine \(k\) that renders \(f(x)\) a valid density function. (b) Find the probability that a random error in measurement is less than \(1 / 2\). (c) For this particular measurement, it is undesirable if the magnitude of the error (i.e., \(|\mathrm{a}:|),\) exceeds 0.8 . What is the probability that this occurs?
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