Question

X is normally distributed if y=3+4X and if E(X)=6 and Var(X)=4 what is E(Y) and the Var(Y) what is the probability of Y<10

Answer 1

\[E(a+bX)=a+bE(X)\]

\[Var(a+bX)=b^2Var(X)\]

Answer 2

okay I got that

Answer 3

If X is normal then Y is also normal

Answer 4

and you apperently already have the mean and variance

Answer 5

yes of x so would that be the same as y?

Answer 6

no

Answer 7

both normal, but they have different means and variances

Answer 8

okay so there both normal, but how do I figure out the expected value of y and the variance only given the values of x?

Answer 9

I gave you the formulas above

Answer 10

y=3+4(6)=42

Answer 11

3+4*6=3+24=27

Answer 12

E[Y]=3+4*E[X] = 3 + 4*6 = 27
V[Y]=4^2V[X] = 16*4 = 64

Answer 13

P(Y<10) = P(3+4X<10) = P(4X<7) = P(X<7/4)

Answer 14

... which is P(X<1.75) = Phi((1.75-6)/2) = Phi(-2.125). This can be found in a table of values for Phi.

Answer 15

I think the whole point of the exercise was to use the information we obtained about Y so we wouldn't need to convert back to X.

Answer 16

It's about 0.0168.

Answer 17

okay the probability formula to me I am only given a t-table and z-table can I figure it out on that table

Answer 18

The z-table is the one you want. You should get 0.0168 at -2.125.

Answer 19

okay yes I get that thank you very much!

Answer 20

in the probability model where did the 7 come from?

Answer 21

You mean where I wrote "P(3+4X<10) = P(4X<7)"? Or somewhere else?

Answer 22

no thats it

Answer 23

Oh. If you take 3+4X < 10 and subtract 3 from both sides, you get 4X < 7.

Answer 24

okay got it and then in the end you divide 7/4 because the var is 4 or because its y=3+4x sorry I i just want to make sure I get it right the numbers are confusing

Answer 25

No problem. It's like this:

We want P(3+4X<10). Looking inside the parentheses, we have:

3 + 4X < 10

Subtract 3 from both sides to get:

4X < 7

Divide both sides by 4 to get:

X < 7/4

Answer 26

okay awesome can you break down the variance formula for me?

Answer 27

Var Y = Var (4X+3)

The +3 part has no effect.

The mult by 4 causes the variance to mult by 4 squared.

So Var Y = 4^2 Var X.
(like b=4 in formula near top of this thread)