Mean And Variance Of Random Variables

To solve this problem, we need to find the mean deviation about the mean for the 6 observations given. Let's break down the solution in a step-by-step manner:

1. Understanding Mean and Variance: The formula for the mean (average) of observations is:

1. Given, the mean of observations is 2:

1. Simplifying, we have:

1. Solving for :

1. Calculating Variance: The formula for variance is:

1. Given that the variance is 23:

1. Simplifying individual terms:

1. Plug in values:

1. Solving Simultaneous Equations: Now, solve for and using:

1. Substitute in equation 2:

1. Solving this:

1. Solving quadratic :

1. So, .
   If , then . If , then . Either way, and take the values 4 and 6.
2. Calculating Mean Deviation: Mean deviation about a mean is given by:

1. Compute mean deviation:

We are given a statistical data set with the mean and the sum of squares as follows:

• Mean of the data: 5.5

The student initially noted two values incorrectly as 4 and 5 instead of 6 and 8. We need to find the variance of the corrected data.

Step-by-Step Solution:

1. Calculate the initial sum of data from the given mean:

1. The incorrect data values are 4 and 5. Their sum is 9.
2. The correct data values are 6 and 8. Their sum is 14.
3. Correct the sum of the data:

1. Calculate the corrected mean:

1. Adjust the sum of squares:
   • Subtract the squares of the incorrect values:

• Add the squares of the correct values:

• Calculate the corrected sum of squares:

1. Calculate the corrected variance:

Therefore, the variance of the corrected data is 7.

The problem requires us to determine the value of , where and are the mean and variance of the transformed observations given by . We are provided with:

• , where
• Variance of is

First, we find the mean of using the first condition:

.

Thus, .

Next, we use the provided variance formula:

.

Since , we have:

.

Now, using , we apply the identity:

.

Substitute for :

.

Solving for :

.

Given , we have .

The transformed observations are . Thus:

.

The mean of is:

.

The variance of is:

.

Finally, calculate :

.

Thus, the answer is .