Respuesta :
Answer:
a) [tex] E(T) = 0.2 E(Y) -1000= 0.2*20000 -1000=3000[/tex]
b) [tex] Sd(T) = \sqrt{0.2^2 Var(Y)}=\sqrt{0.2^2 8000^2}= 1600[/tex]
c) Assuming 20 million of families and each one with a mean of income of 20000 for each family approximately then total income would be:
[tex] E(T) = 20000000*20000= 40000 millions[/tex]
And if we replace into the formula of T we have:
[tex] T = 0.2*400000x10^6 -1000= 790000 millions[/tex]
Approximately.
Step-by-step explanation:
For this case we knwo that Y represenet the random variable "Income" and we have the following properties:
[tex] E(Y) = 20000, Sd(Y) = 8000[/tex]
We define a new random variable T "who represent the taxes"
[tex] T = 0.2(Y-5000) = 0.2Y -1000[/tex]
Part a
For this case we need to apply properties of expected value and we have this:
[tex] E(T) = E(0.2 Y -1000)[/tex]
We can distribute the expected value like this:
[tex] E(T) = E(0.2 Y) -E(1000)[/tex]
We can take the 0.2 as a factor since is a constant and the expected value of a constant is the same constant.
[tex] E(T) = 0.2 E(Y) -1000= 0.2*20000 -1000=3000[/tex]
Part b
For this case we need to first find the variance of T we need to remember that if a is a constant and X a random variable [tex] Var(aX) = a^2 Var(X)[/tex]
[tex] Var(T) = Var (0.2Y -1000)[/tex]
[tex] Var(T)= Var(0.2Y) -Var(1000) + 2 Cov(0.2Y, -1000)[/tex]
The covariance between a random variable and a constant is 0 and a constant not have variance so then we have this:
[tex] Var(T) =0.2^2 Var(Y)[/tex]
And the deviation would be:
[tex] Sd(T) = \sqrt{0.2^2 Var(Y)}=\sqrt{0.2^2 8000^2}= 1600[/tex]
Part c
Assuming 20 million of families and each one with a mean of income of 20000 for each family approximately then total income would be:
[tex] E(T) = 20000000*20000= 40000 millions[/tex]
And if we replace into the formula of T we have:
[tex] T = 0.2*400000x10^6 -1000= 790000 millions[/tex]
Approximately.
