Question

suppose x is a random variable with mean 20 and standard deviation 5. also suppose that Y is a random variable with mean 40 and standard deviation 10. find the variance and standard deviation of the random variable z

A) Z = 2 + 10x

B) Z = X + Y

Answer 1

I have no idea how to start this

Answer 2

Okay, let's start with notations.
\[\Large \mathbb{E}\] means "mean"

\[\Large \text{Var}\] means Variance.

It would help you to know that \[\Large \mathbb{E} [X + Y]=\mathbb{E}[X]+\mathbb{E}[Y]\] and that \[\Large \mathbb E [kX]=k\mathbb E [X]\]

Answer 3

Also, for Variances, we have
\[\Large \text{Var}[X+Y] = \text{Var}[X] + \text{Var}[Y]\]
and \[\Large \text{Var}[kX] = k^{\color{red}2}\text{Var}[X]\]
Note the exponent.

Answer 4

what will the k equal in this case

Answer 5

k is any constant.
Now, here's the deal, I'll demonstrate the first item to you, then you'll demonstrate the second item to me. Sound fair?

Answer 6

yes

Answer 7

Okay, let's begin.
\[\Large Z = 2 + 10X\]We need its mean. So...
\[\Large \mathbb E [Z] = \mathbb E [2 + 10X]\]
By the first property of means, we get
\[\Large \mathbb E[Z] = \mathbb E[2] + \mathbb E [10X]\]

Answer 8

Now the mean of a constant is itself, and the variance of a constant is zero (rightly so, constants don't CHANGE, hence the name)
So...
\[\Large \mathbb E [Z] = \color{red}2 + \color{blue}{10}\mathbb E[X]\] by the second property of means ( the one that says E[kX] = k E[X]
And we're almost done here, since we know that the mean of X is 20. We can just replace E[X] with 20.

\[\Large \mathbb E[Z] = 2 + 10(\color{green}{20})\]

\[\Large \mathbb E[Z] = 2 + 200 = 202\]
And that's your mean.

Answer 9

Questions?

Answer 10

so will the SD affect anything? or is it as simple as that

Answer 11

Not yet, we have only ever tried to find the mean. Now we find the standard deviation.
Let's do that by getting the variance, first.
\[\Large Z = 2 + 10X\]
\[\Large \text{Var}[Z] = \text{Var}[2+10X]\]

By the first property of Variances, we get
\[\Large \text{Var}[Z] = \text{Var}[2] + \text{Var}[10X]\]

Answer 12

Now, the Variance of 2... a constant, since 2 doesn't VARY at all, is zero. So we are left with
\[\Large \text{Var}[Z] = \text{Var}[10X]\]

By the second property of Variances, 10 can move out, but becomes squared in the process:
\[\Large \text{Var}[Z] = 10^{\color{red}2}\text{Var}[X] = 100\text{Var}[X]\]

Answer 13

Now, since the standard deviation of X is 5, it follows that its variance is 25, since standard deviation is the square root of the variance.
\[\Large \text{Var}[Z] = 100(25)\]

Answer 14

wasnt the sd 5? so the varience would be 25

Answer 15

ahh

Answer 16

That's right :)

Answer 17

Now, all that's left is to simplify
\[\Large \text{Var}[Z] = 2500\]
The Variance of Z is 2500, so to get the standard deviation, simply get the square root of the variance, and you'll get the SD of Z is 50

Answer 18

So Z has a var of 2500 and a sd of 50

Answer 19

ah beat me to it

Answer 20

yup. So much for the first item. Now do the second one :)

Answer 21

LOL We agreed that *I* was to do the first item and you do the second :D

Answer 22

so for B, it is Z = (20+40) Z has a mean of 60

Answer 23

then Z = 60(5) + 60(10) or is it 60(5+10)

Answer 24

Correct. The SD?

Answer 25

No... it's not that simple :P
Get their variances :D

Answer 26

ok, give me a min

Answer 27

so the Var is 900 (Z)

Answer 28

30?

Answer 29

now i have another one, Z=10x-2 is it the same answer as A)?

Answer 30

also is the first part why didnt we multiply the mean (202) by the variance? like in the second question

Answer 31

Are X and Y are independent?