Question

Which math example correspond to which property of the expected value below?

Properties of expected value:

1. E[c] = c

2. E[c · X] = c · E[X]

3. E[X + Y ] = E[X] + E[Y ]

4. E[aX + b] = aE[X] + b

• Your sample data set is: {3, 3, 3, 3}. Calculate The average! What is your result? if you take the average of {5, 5, 5} or the data set {−1, −1, −1, −1, −1}, What would you get? (Try to guess, instead of calculating. Observe a rule.) Which property does your insight correspond to (give the number)?

• Your sample data set contains variable X: X = {2, 5, 1, 6} – Take the average of your data. Then multiply the mean by −2, and add 4 to that number. Write down these operations in 1 line mathematically, using our notation. What is your result? – Multiply each and every value of the data set by −2, and add 4 to each number you got. This new, X˜ variable - how does it relate to X? Take the average of X˜. What is your result? Which property does your insight correspond to from above (give the number)?

• Your sample data set contains variable X: X = {2, 5, 1, 6} 2 – Take the average of your data. Then multiply the mean by 3. Write down these operations in 1 line mathematically, using our notation. What is your result? – Multiply each and every value of the data set by 3. This new, X˜ variable - how does it relate to X? Take the average of X˜. What is your result? Which property does your insight correspond to from above (give the number)?

• You have 2 variables in your sampled data: X = {1, 8, 4} and Y = {3, 7, 1}. – Take the average of X. Take the average of Y . Now sum these averages. Write down these operations in 1 line with math notation. What do you get? – For every observation, add up X and Y . You will get a new variable (3 numbers), which we denote by X + Y . Take average of X + Y . What do you get? Which property does your insight correspond to from above (give the number)?

Answer 1

1) We can observe that the average of the data set (3,3,3,3) and (5,5,5) and ( -1,-1,-1,-1) is 3, 5 and -1 respectively.
It relates to property 1 mentioned above.

2) ( ( 2 + 5 + 1 + 6 ) / 4 ) * ( -2 ) + 4 = 3

( 2 * -2 + 4 ) = 0
( 5 * -2 + 4 ) = -6
( 1 * -2 + 4 ) = 2
( 6 * -2 + 4 ) = 8

Average of these new variables is 1
( 0 - 6 + 2 + 8 ) / 4 = 4 / 4 = 1

This is related to property 4

3) ( ( 2 + 5 + 1 + 6 ) / 4 ) * ( 3 ) = 3.5 * 3 = 10.5

2 * 3 = 6
5 * 3 = 15
1 * 3 = 3
6 * 3 = 9

( 6 + 15 + 3 + 9 ) / 4 = 33 / 4 = 8.25

It is related to mathematical property 2

4) ( ( 1 + 8 + 4 ) / 3 ) + ( ( 3 + 7 + 1) / 3 ) = 8

( ( 1 + 3 ) + ( 8 + 7 ) + ( 4 + 1 ) ) / 3 = 24 / 3 = 8

It is related to mathematical property 3