Author Topic: 'Special Properties' e.g. E(aX + b) = aE(X) + b (Read 6558 times) Tweet Share

sxcalexc · « **on:** November 09, 2008, 05:34:25 pm »

As per the title. Does anyone have any example problems using this 'formula'? I have got it written down but have never had to use or have never seen a question applying it. Also I have seen Var(X + Y) = Var(X) + Var(Y). Can anyone clarify what these signify? Cheers

BiG DaN · « **Reply #1 on:** November 09, 2008, 05:37:18 pm »

simple example

if E(X)=5
what does E(2X+1) equal

=2E(X) + 1
=11

shinny · « **Reply #2 on:** November 09, 2008, 05:39:48 pm »

Here's one. It's from the MHS 2007 Methods SAC, and only other place I've seen these is the Essential methods book, so don't fret if you can't do it. I don't think I've seen anything like this on a trial or VCAA before so yeh, don't freak out if you get stuck.

1. The random variable L has mean 65 and variance 16. The random variable S has mean 50 and variance 9. Variables L and S are independent and the random variable D is given by $D=4L-5S$
a. i. Find the mean of D
ii. Find the variance of D
b. The weights of large eggs are normally distributed with mean 65 grams and standard deviation of 4 grams. The weights of standard eggs are normally distributed with mean 50 grams and a standard deviation of 3 grams. One large egg and one standard egg are chosen at random. Find the probability that the weight of the standard egg is more than $\frac{4}{5}$ of the weight of the large egg.

EDIT: Just ask if anyone wants solutions and I'll chuck 'em up later.

Collin Li · « **Reply #3 on:** November 09, 2008, 05:45:55 pm »

Quote from: sxcalexc on November 09, 2008, 05:34:25 pm

As per the title. Does anyone have any example problems using this 'formula'? I have got it written down but have never had to use or have never seen a question applying it. Also I have seen Var(X + Y) = Var(X) + Var(Y). Can anyone clarify what these signify? Cheers

You should know that the expectation operator is linear (the formula in the topic), but you do not need to know about Var(X + Y) = Var(X) + Var(Y).

sxcalexc · « **Reply #4 on:** November 09, 2008, 05:50:28 pm »

Quote from: coblin on November 09, 2008, 05:45:55 pm

Quote from: sxcalexc on November 09, 2008, 05:34:25 pm
As per the title. Does anyone have any example problems using this 'formula'? I have got it written down but have never had to use or have never seen a question applying it. Also I have seen Var(X + Y) = Var(X) + Var(Y). Can anyone clarify what these signify? Cheers

You should know that the expectation operator is linear (the formula in the topic), but you do not need to know about Var(X + Y) = Var(X) + Var(Y).

Oh sweet, are you implying we don't have to know how to apply it?

shinny · « **Reply #5 on:** November 09, 2008, 05:52:56 pm »

Quote from: sxcalexc on November 09, 2008, 05:50:28 pm

Quote from: coblin on November 09, 2008, 05:45:55 pm
Quote from: sxcalexc on November 09, 2008, 05:34:25 pm
As per the title. Does anyone have any example problems using this 'formula'? I have got it written down but have never had to use or have never seen a question applying it. Also I have seen Var(X + Y) = Var(X) + Var(Y). Can anyone clarify what these signify? Cheers

You should know that the expectation operator is linear (the formula in the topic), but you do not need to know about Var(X + Y) = Var(X) + Var(Y).
Oh sweet, are you implying we don't have to know how to apply it?

They could just state it on the exam paper, and then you'll have to apply it. I saw one trial paper do that.

bec · « **Reply #6 on:** November 09, 2008, 06:20:11 pm »

Shinjitsuzx, is this right?

1. The random variable L has mean 65 and variance 16. The random variable S has mean 50 and variance 9. Variables L and S are independent and the random variable D is given by $D=4L-5S$
a. i. Find the mean of D
E(D) =4E(L) -5E(S)
= 4*65 - 5*50
= 10

ii. Find the variance of D
Var(D) = Var(4L) -Var(5S)
~~= 16*16 - 25*9~~ 16*16 + 25*9
~~= 31~~ = 481

b. The weights of large eggs are normally distributed with mean 65 grams and standard deviation of 4 grams. The weights of standard eggs are normally distributed with mean 50 grams and a standard deviation of 3 grams. One large egg and one standard egg are chosen at random. Find the probability that the weight of the standard egg is more than $\frac{4}{5}$ of the weight of the large egg.

Let L be the mass of a large egg, let S be the mass of a standard egg
E(L)=65; SD(L)=4; Var(L)=16
E(S)=50; SD(S)=3; Var(S)=9

$Pr(S > \frac {4}{5} L) = Pr(S- \frac {4}{5} L > 0)$
Assume $S-\frac {4}{5}L$ to be a normal distribution with:
$E(S-\frac {4}{5}L)=E(S)-\frac {4}{5}E(L)$
$= 50- \frac {4}{5}*65 = -2$

$Var(S-\frac {4}{5}L)=Var(S)+ \frac {4^2}{5^2}Var(L)$
$= \frac {369}{25}$

$Pr(S- \frac {4}{5} L > 0) = 0.301$

(edit: changed solution...which is still wrong)

sxcalexc · « **Reply #7 on:** November 09, 2008, 06:20:42 pm »

Quote from: shinjitsuzx on November 09, 2008, 05:52:56 pm

Quote from: sxcalexc on November 09, 2008, 05:50:28 pm
Quote from: coblin on November 09, 2008, 05:45:55 pm
Quote from: sxcalexc on November 09, 2008, 05:34:25 pm
As per the title. Does anyone have any example problems using this 'formula'? I have got it written down but have never had to use or have never seen a question applying it. Also I have seen Var(X + Y) = Var(X) + Var(Y). Can anyone clarify what these signify? Cheers

You should know that the expectation operator is linear (the formula in the topic), but you do not need to know about Var(X + Y) = Var(X) + Var(Y).
Oh sweet, are you implying we don't have to know how to apply it?

They could just state it on the exam paper, and then you'll have to apply it. I saw one trial paper do that.

Ah, right, did you have solutions to your posted question?

vce08 · « **Reply #8 on:** November 09, 2008, 06:22:32 pm »

Quote from: bec on November 09, 2008, 06:20:11 pm

ii. Find the variance of D
Var(D) = Var(4L) -Var(5S)
= 16*16 - 25*9
= 31

That step is wrong I think
instead of the - sign
use a + sign

bec · « **Reply #9 on:** November 09, 2008, 06:25:10 pm »

Yep, you're right. I always make that mistake! Thanks vce08

shinny · « **Reply #10 on:** November 09, 2008, 06:25:50 pm »

Yeh its a plus sign there. Look at it this way:
$Var(D)=Var(4L-5S)$
$=Var(4L+(-5S))$
$=4^{2}Var(L)+(-5)^{2}Var(S)$
$=16Var(L)+25Var(S)$

Collin Li · « **Reply #11 on:** November 09, 2008, 06:43:51 pm »

You don't need to know the Var(X+Y) = Var(X) + Var(Y) rule.

You only need to know the rules concerning 1 random variable (just X, not both X and Y).

$\mbox{E}(aX+b) = a\mbox{E}(X) + b$

$\mbox{Var}(aX+b) = a^2\mbox{Var}(X)$

And yes, you need to apply them.

shinny · « **Reply #12 on:** November 09, 2008, 06:49:59 pm »

OK since someone requested:

1.a.i. $E(D)=E(4L-5S)$

$=E(4L)-E(5S)$

$=4E(L)-5E(S)$

$=(4)(65)-(5)(50)$

$=260-250$

$=10$
ii. $Var(D)=Var(4L+(-5)S)$

$=Var(4L)+Var((-5)S)$

$=4^{2}Var(L)+(-5)^{2}Var(S)$

$=(16)(16)+(25)(9)$

$=256+225$

$=481$
b. First define your variables. The letters in the first part are a clue.
Let L be the weight of a large egg
Let S be the weight of a standard egg
Next, turn what they're asking into a mathematical formula. Just disect the question word by word and this shouldn't be hard. Probability indicates that obviously, you're finding a probability, so you'll need Pr(something). Next, since it's asking for the probability that a standard egg is greater than four-fifths of the weight of the large egg, using your variables, this reads as $S>\frac{4}{5}L$ , so combine the two to get $Pr(S>\frac{4}{5}L)$
So working from there, you're actually allowed to move terms inside the Pr, so multiply both sides by 5 and transpose to get:
$Pr(5S-4L>0)$
$=Pr(4L-5S<0)$
$=Pr(D<0)$ <---recognise the link between this question and the first one
Then using the values you worked out before, use the normal distribution to calculate it.
$=Pr(Z<\frac{0-10}{\sqrt{481}})$
$\approx 0.324$

bec · « **Reply #13 on:** November 09, 2008, 07:10:39 pm »

I know I used a different (clumsier) method from shinjitsuzx but what's wrong with my way?
Have a made a silly mistake or is my method just wrong?

Quote from: bec on November 09, 2008, 06:20:11 pm

Let L be the mass of a large egg, let S be the mass of a standard egg
E(L)=65; SD(L)=4; Var(L)=16
E(S)=50; SD(S)=3; Var(S)=9

$Pr(S > \frac {4}{5} L) = Pr(S- \frac {4}{5} L > 0)$
Assume $S-\frac {4}{5}L$ to be a normal distribution with:
$E(S-\frac {4}{5}L)=E(S)-\frac {4}{5}E(L)$
$= 50- \frac {4}{5}*65 = -2$

$Var(S-\frac {4}{5}L)=Var(S)+ \frac {4^2}{5^2}Var(L)$
$= \frac {369}{25}$

$Pr(S- \frac {4}{5} L > 0) = 0.301$

(edit: changed solution...which is still wrong)

onlyfknhuman · « **Reply #14 on:** November 09, 2008, 07:10:49 pm »

Quote from: shinjitsuzx on November 09, 2008, 06:49:59 pm

OK since someone requested:

1.a.i. $E(D)=E(4L-5S)$

$=E(4L)-E(5S)$

$=4E(L)-5E(S)$

$=(4)(65)-(5)(50)$

$=260-250$

$=10$
ii. $Var(D)=Var(4L+(-5)S)$

$=Var(4L)+Var((-5)S)$

$=4^{2}Var(L)+(-5)^{2}Var(S)$

$=(16)(16)+(25)(9)$

$=256+225$

$=481$
b. First define your variables. The letters in the first part are a clue.
Let L be the weight of a large egg
Let S be the weight of a standard egg
Next, turn what they're asking into a mathematical formula. Just disect the question word by word and this shouldn't be hard. Probability indicates that obviously, you're finding a probability, so you'll need Pr(something). Next, since it's asking for the probability that a standard egg is greater than four-fifths of the weight of the large egg, using your variables, this reads as $S>\frac{4}{5}L$ , so combine the two to get $Pr(S>\frac{4}{5}L)$
So working from there, you're actually allowed to move terms inside the Pr, so multiply both sides by 5 and transpose to get:
$Pr(5S-4L>0)$
$=Pr(4L-5S<0)$
$=Pr(D<0)$ <---recognise the link between this question and the first one
Then using the values you worked out before, use the normal distribution to calculate it.
$=Pr(Z<\frac{0-10}{\sqrt{481}})$
$\approx 0.324$

im freaking out :S

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Author Topic: 'Special Properties' e.g. E(aX + b) = aE(X) + b (Read 6558 times) Tweet Share

sxcalexc

'Special Properties' e.g. E(aX + b) = aE(X) + b

BiG DaN

Re: 'Special Properties' e.g. E(aX + b) = aE(X) + b

shinny

Re: 'Special Properties' e.g. E(aX + b) = aE(X) + b

Collin Li

Re: 'Special Properties' e.g. E(aX + b) = aE(X) + b

sxcalexc

Re: 'Special Properties' e.g. E(aX + b) = aE(X) + b

shinny

Re: 'Special Properties' e.g. E(aX + b) = aE(X) + b

bec

Re: 'Special Properties' e.g. E(aX + b) = aE(X) + b

sxcalexc

Re: 'Special Properties' e.g. E(aX + b) = aE(X) + b

vce08

Re: 'Special Properties' e.g. E(aX + b) = aE(X) + b

bec

Re: 'Special Properties' e.g. E(aX + b) = aE(X) + b

shinny

Re: 'Special Properties' e.g. E(aX + b) = aE(X) + b

Collin Li

Re: 'Special Properties' e.g. E(aX + b) = aE(X) + b

shinny

Re: 'Special Properties' e.g. E(aX + b) = aE(X) + b

bec

Re: 'Special Properties' e.g. E(aX + b) = aE(X) + b

onlyfknhuman

Re: 'Special Properties' e.g. E(aX + b) = aE(X) + b

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