Statistics - variance question

In summary, the conversation discusses an intro course test question involving random variables X and Y with given values for p(X,Y), Var(X), and Var(Y). The question asks for the computation of Var(X-2Y) using the formula Var(X-Y) = Var(X) + Var(Y) - 2Cov(X,Y) since X and Y are not independent. The person asking for help is unsure how to proceed with the given information.
  • #1
hanelliot
18
0
Studying for an intro course test and I have no one to compare it to right now.. any help would be appreciated.

Here is the question.

Q. Suppose X and Y are random variables such that p(X,Y)=1/3, Var(X) = 9 and Var(Y) = 1. Compute Var(X-2Y).

Since X and Y are not independent, we are using this formula: Var(X-Y) = Var(X) + Var(Y) - 2Cov(X,Y), correct?
So, Var(X-2Y) = Var(X) - 4Var(Y) - 2Cov(X,Y)? How do I proceed from here?
 
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  • #2
Hi, you may want to clarify what exactly p(X,Y) means here.
 
  • #3


I understand that studying for an intro course test can be challenging, especially when you don't have anyone to compare your work to. However, I am happy to assist you with this question on variance.

First, you are correct in using the formula Var(X-Y) = Var(X) + Var(Y) - 2Cov(X,Y) since X and Y are not independent. To proceed, we need to calculate the covariance (Cov) of X and Y.

Cov(X,Y) = E[(X-E[X])(Y-E[Y])] where E[X] and E[Y] are the expected values of X and Y respectively. Since p(X,Y) = 1/3, we can calculate the expected values as follows:

E[X] = 1*(1/3) + 2*(1/3) + 3*(1/3) = 2
E[Y] = 1*(1/3) + 2*(1/3) + 3*(1/3) = 2

Now, we can calculate the covariance using the formula:

Cov(X,Y) = E[(X-2)(Y-2)] = E[XY - 2X - 2Y + 4] = E[XY] - 4E[X] - 4E[Y] + 16

To calculate E[XY], we can use the fact that X and Y are independent, so E[XY] = E[X]E[Y] = 4. Therefore, Cov(X,Y) = 4 - 4(2) - 4(2) + 16 = 4.

Now, we can plug this value into the formula for Var(X-2Y):

Var(X-2Y) = Var(X) - 4Var(Y) - 2Cov(X,Y) = 9 - 4(1) - 2(4) = 1

Therefore, the variance of X-2Y is 1. I hope this helps and best of luck on your test! Remember to always check your calculations and seek help if needed.
 

1. What is variance in statistics?

Variance is a measure of how spread out a set of data points are from the average. It measures the average squared distance of each data point from the mean of the data set.

2. How is variance calculated?

Variance is calculated by taking the sum of the squared differences between each data point and the mean, and then dividing that sum by the total number of data points. This can also be represented by the formula (sum of (x - mean)^2) / n, where x represents the data points and n represents the total number of data points.

3. What does a high variance indicate?

A high variance indicates that the data points are spread out over a wide range from the mean. This can suggest that the data is more diverse or has a larger range of values.

4. What does a low variance indicate?

A low variance indicates that the data points are clustered closer to the mean. This can suggest that the data is more consistent or has a smaller range of values.

5. How is variance used in statistics?

Variance is used in statistics to measure the variability of a data set and to understand the spread of the data points. It is also used in calculating other important statistical measures such as standard deviation and correlation.

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