1. ## Question about mean and variance

Question : If $X_1$ has mean 4 and variance 9 while $X_2$ has mean -2 and variance 5, and the two are independent, determine
i) $E(2X_1 + X_2 -5)$
ii) $Var(2X_1 + X_2 -5)$

i) $E(2X_1 + X_2 -5)$ = $2[E(X_1)] + [E(X_2) -5]$ = $8 + [-2 -5]$ = $1$

ii) $Var(2X_1 + X_2 -5)$ = $2[Var(X_1)] + [Var(X_2) -5]$ = $2(9) + [5 -5]$ = $18 + 0$ = $18$

Is my answer correct and is my method for solving the answer crorect or no

2. Scalars a and b acting on a random variable X as:

$aX+b$

and act on a random variables variance as:

$a^2\sigma^2_x$;

Why? Think about it this way. If you had the ages of 10 people, and you multiplied their ages by 2 and added 10 years, the 10 years isn't going to affect the variation because they have all been adjusted evenly by 10 years. What will affect their ages is multiplying by the 2, and the variance is adjust by the square of the scalar.

3. Originally Posted by zorro
Question : If $X_1$ has mean 4 and variance 9 while $X_2$ has mean -2 and variance 5, and the two are independent, determine
i) $E(2X_1 + X_2 -5)$
ii) $Var(2X_1 + X_2 -5)$

i) $E(2X_1 + X_2 -5)$ = $2[E(X_1)] + [E(X_2) -5]$ = $8 + [-2 -5]$ = $1$

ii) $Var(2X_1 + X_2 -5)$ = $2[Var(X_1)] + [Var(X_2) -5]$ = $2(9) + [5 -5]$ = $18 + 0$ = $18$

Is my answer correct and is my method for solving the answer crorect or no
$E(aX_1 + bX_2) = a E(X_1) + b E(X_2)$ and $E(cX + d) = c E(X) + d$ and so your answer to (i) is correct.

However, if $X_1$ and $X_2$ are independent then $Var(aX_1 + bX_2) = a^2 Var(X_1) + b^2 Var(X_2)$ and $Var(cX + d) = c^2 Var(X)$. Therefore your answer to (ii) is wrong.

4. ## Is this correct?

Originally Posted by mr fantastic
$E(aX_1 + bX_2) = a E(X_1) + b E(X_2)$ and $E(cX + d) = c E(X) + d$ and so your answer to (i) is correct.

However, if $X_1$ and $X_2$ are independent then $Var(aX_1 + bX_2) = a^2 Var(X_1) + b^2 Var(X_2)$ and $Var(cX + d) = c^2 Var(X)$. Therefore your answer to (ii) is wrong.

Mr fantastic is this correct

$Var(2X_1 + X_2 - 5)$ = $2^2 [Var(X_1)] + Var(X_2) - 5$ = $4 * 9 + 5 -5$ = $36$ .......Is this correct

5. Originally Posted by zorro
Mr fantastic is this correct

$Var(2X_1 + X_2 - 5)$ = $2^2 [Var(X_1)] + Var(X_2) - 5$ = $4 * 9 + 5 -5$ = $36$ .......Is this correct
Not quite.

$Var(X_2 - 5) = Var(X_2)$ (using $Var(cX + d) = c^2 Var(X)$ and noting that c = 1 and d = -5). Therefore $Var(2 X_1 + X_2 - 5) = 2^2 Var(X_1) + Var(X_2 - 5) = 2^2 Var(X_1) + Var(X_2) = ....$.

6. ## Is this correct?

Originally Posted by mr fantastic
Not quite.

$Var(X_2 - 5) = Var(X_2)$ (using $Var(cX + d) = c^2 Var(X)$ and noting that c = 1 and d = -5). Therefore $Var(2 X_1 + X_2 - 5) = 2^2 Var(X_1) + Var(X_2 - 5) = 2^2 Var(X_1) + Var(X_2) = ....$.

7. Originally Posted by zorro