Chapter 9: Problem 3
Let
Short Answer
Expert verified
The constants that make and independent are .
Step by step solution
01
Define Variables
Let's define the given variables first. It is given that , , , are random samples from a normal distribution that is . and are also defined.
02
Calculate Covariance
The first step is to calculate the covariance of and . For the variables to be independent, their covariance should be zero. Using the equation for the calculation of covariance: COV( , ) = E[ ] - E[ ]E[ ]. Because are distributed N(0, ), then E[ ] = 0 for any n. So E[ ] = E[ ] = 0. Then COV( , ) = E[ ] becomes our new equation.
03
Expression for Expected Value of Y*Q
Now, let's find the expression for E[ ]. By multiplying and and taking the expected value: E[ ] = E[ ]. This simplifies to: E[ ] = E[ ] + E[ ] - E[ ] - E[ ].
04
Simplify the Expression
The next step is to simplify the expected value expression obtained above. Because are independent, the expected value of their product is the product of their expected values. As E[ ] = 0 for any n, E[ ] = E[ ] = 0. Now our E[ ] becomes E[ ] + E[ ].
05
Independence of Variables
For and to be independent, the covariance of these two variables should be zero. So COV( , ) = E[ ] = 0. Setting the expression E[ ] + E[ ] to be zero gives = - . However, the choice of and is arbitary, thus, any pair of and should hold = - . Hence, .
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Covariance
Covariance is a measure of how two random variables change together. In simpler terms, it shows the relationship between two variables, helping us understand if they tend to increase or decrease simultaneously. If the covariance is positive, it means that as one variable increases, the other tends to increase as well. Conversely, a negative covariance indicates that as one goes up, the other tends to go down.
To find the covariance between two variables, we use the formula: and being zero is key. It implies these two variables don't influence each other's fluctuations and suggests independence.
This is why it's crucial to simplify our calculations to find when , helping us derive relations like and extending this idea to find the solution for and .
To find the covariance between two variables, we use the formula:
-
- This can also be rephrased as
for simplified calculation
This is why it's crucial to simplify our calculations to find when
Independence
Independence in probability and statistics refers to the idea that two random variables do not affect each other's outcomes. Simply put, whether one variable occurs or not has no impact on the probability of the other variable occurring.
In mathematical terms, two events and are independent if the probability of both events occurring together is the same as the product of their separate probabilities: and as this is given as a condition in the problem. Given that , this statistical property helps affirm that the variables are independent if they follow normal distribution settings. The conclusion derived from the independence condition also conforms with calculating zero covariance, which reinforces their independent status under these conditions.
Thus, when you find that the covariance is zero, like in our task, it reinforces potential independence, especially in the context of normal distribution.
In mathematical terms, two events
Thus, when you find that the covariance is zero, like in our task, it reinforces potential independence, especially in the context of normal distribution.
Normal Distribution
A normal distribution, also known as the Gaussian distribution, is one of the most important probability distributions in statistics. This distribution is characterized by its bell-shaped curve, where most of the data points cluster around a central mean, and the probabilities gradually decrease as you move towards the extremes or tails.
Key properties of a normal distribution include: are distributed as . This implies they follow a normal distribution centered around zero with variance .
Understanding that and involve combinations of these normally distributed variables helps derive insights about the characteristics of and . Knowing they are extracted from a normal distribution makes solving the problem more intuitive, leading to the independence and covariance concepts utilized in the solution.
Key properties of a normal distribution include:
- The mean, median, and mode of the distribution are equal.
- It is symmetric about the mean.
- Approximately 68% of the data lies within one standard deviation (
) of the mean, 95% within two, and 99.7% within three.
Understanding that