A Introduction to Statistics Question on Normally distributed RVs

In summary, we can determine that Z has a normal distribution with mean 1 and variance 10. The covariance between X and Z is 6, and the correlation between X and Z is approximately 1.89. These calculations are based on the properties of normal random variables and the independence of X and Y.
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CaptainBlack
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Question from Jim on Yahoo Answers:

Let X and Y be independent random variables with X ~ N(3, 1)
and Y ~ N(6, 9). Put Z = Y − 4X + 7.

(a) State the distribution of Z.
(b) Calculate Cov(X, Z).
(c) Calculate corr(X, Z)

The things you should know are:

1. The sum of normal random variables is a normal random variable with mean equal to the sum of the means and variance equal to the sum of the variances.

2. Multiplying a normal RV by a constant gives a normal RV with mean equal to the constant times the mean of the original RV and variance equal to the square of the constant times the variance of the original RV

3. Adding a constant to a normal RV gives you a RV with mean equal to the original plus the constant and variance unchanged.

(a) So applying these to \(Z\) you find \(Z \sim N( (6-12+7), (1+9) ) = N(1,10) \)

For (b) you need to write:

\[Cov(X,Z)= E( (X-\mu_X)(Z-\mu_Z) )= E(X*Z - \mu_X Z-\mu_Z X +\mu_X \mu_Z)=E(X*Z) - \mu_X \mu_Z\]

Now you expand the remaining expectation above replacing \(Z\) by \(Y- 4X+7\) using the linearity properties of the expectation operator and the independence of \(X\) and \(Y\)

(c) is a trival application of the definition once you have done (a) and (b)
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(c) Using the definition of correlation, we have:

\[corr(X,Z) = \frac{Cov(X,Z)}{\sqrt{Var(X) Var(Z)}} = \frac{E(X*Z) - \mu_X \mu_Z}{\sqrt{Var(X) Var(Z)}}\]

Plugging in the values calculated in (a) and (b), we get:

\[corr(X,Z) = \frac{E(X(Y-4X+7)) - 3*1}{\sqrt{1 * 10}} = \frac{E(XY) - 12}{\sqrt{10}}\]

To calculate \(E(XY)\), we can use the independence of \(X\) and \(Y\) and the linearity property of the expectation operator to get:

\[E(XY) = E(X) * E(Y) = 3 * 6 = 18\]

Thus, we have:

\[corr(X,Z) = \frac{18 - 12}{\sqrt{10}} = \frac{6}{\sqrt{10}} \approx 1.89\]
 

FAQ: A Introduction to Statistics Question on Normally distributed RVs

What is a normally distributed random variable (RV)?

A normally distributed RV is a type of continuous random variable that follows a bell-shaped curve when graphed. It is commonly used to represent real-world data that is symmetric and exhibits a central tendency.

How is a normally distributed RV different from other types of RVs?

A normally distributed RV is different from other types of RVs because it follows a specific probability distribution called the normal distribution. This distribution is characterized by its mean and standard deviation, which determine the shape and spread of the bell curve.

How do you calculate probabilities for a normally distributed RV?

To calculate probabilities for a normally distributed RV, you can use the standard normal distribution table or a statistical calculator. You will need to know the mean and standard deviation of the RV, as well as the specific value or range of values you are interested in finding the probability for.

What is the central limit theorem and how does it relate to normally distributed RVs?

The central limit theorem states that as the sample size increases, the sampling distribution of the sample means will approach a normal distribution regardless of the shape of the population distribution. This means that normally distributed RVs are important in statistical analysis because many real-world data sets can be approximated by a normal distribution, making it easier to make statistical inferences.

How are normally distributed RVs used in hypothesis testing?

Normally distributed RVs are used in hypothesis testing to determine whether there is a significant difference between two groups or if an observed result is due to chance. This is done by calculating the probability of obtaining the observed result under the assumption that the null hypothesis is true. If this probability is low enough, the null hypothesis is rejected in favor of the alternative hypothesis.

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