help with estimation stats cant get answer on internet

[glass bane]

kindly help, i cant find the answer on internet at all.

[abstract veldt]

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[still sigil]

For multiple independent normally distributed variables

You should have seen something in your lecture regarding their sum

[glass bane]

my teacher didnt discuss this topic

yet he gave an assignment on this

[still sigil]

Then now is a good time to talk about it

[glass bane]

okayy

so i did try but the variance is coming out negative?

[still sigil]

The variance is still the sum of the variances

[glass bane]

but acc to the X given

wont we use that?

[still sigil]

So just to be clear

[glass bane]

like the to mean is 54 for the given X

[glass bane]

youre saying it'd be 45?

[still sigil]

Well, hold on before we do anything

[glass bane]

okay

[still sigil]

Basically, my statement is as follows

Let $Y_1 \sim \mathcal{N}(\mu_1, \sigma_1^2), ..., Y_n \sim \mathcal{N}(\mu_n, \sigma_n^2)$

[dense veldt]

Rion

[still sigil]

Then: $Y_1 + ... + Y_n \sim \mathcal{N}(\mu_1 + ... + \mu_n, \sigma_1^2 + ... + \sigma_n^2)$

[dense veldt]

Rion

[glass bane]

😕 huh

[still sigil]

(when Y_1, ..., Y_n are mutually independent)

For the proof, I refer you to this, which you can read now or later

https://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables

But basically, we can exploit this property

By selecting the right Yi

[glass bane]

okayy uh

[still sigil]

My claim is as follows:
$Y_1 = 0.3 X_1$, $Y_2 = -X_2$, $Y_3 = 1.7 X_3$ are independent normally distribued random variables

[dense veldt]

Rion

[still sigil]

If you agree with me, do determine their parameters

[glass bane]

yes

so yes i tried. the variance came out negative

and the mean was 54

[still sigil]

What variance?

For now, please tell me the mean and variance of Y1, Y2 and Y3

[glass bane]

mean y1 is 9
y2 is -40
y3 is 85

variance is y1 4.5, y2 is -25 and y3 is 8.5

[still sigil]

For the means, I agree

for the variances, I don't

[glass bane]

youre saying it'll be the same

[still sigil]

No

[glass bane]

then?

[still sigil]

remember, for a rv X, then $Var(aX) = a^2 Var(X)$

[dense veldt]

Rion

[glass bane]

OH

[still sigil]

So that explains a lot of problems you were having

[glass bane]

☠️ i did see this but in that question it was just x1 +x2

so i forgot that

[glass bane]

lemme try

[still sigil]

Again, tell me the respective variances of Y1, Y2 and Y3

[glass bane]

y1 0.135
y2 25
y3 14.45

?

[still sigil]

I don't have the same thing for Var(Y1)

[glass bane]

aye

uh

sorry

1.35

[still sigil]

Yeah, correct

V(Y1)=1.35
V(Y2)=25
V(Y3)=14.45

Now, we can proceed to compute the distribution of $X = Y_1 + Y_2 + Y_3$

[dense veldt]

Rion

[still sigil]

We already know it is a normal distribution: what are its parameters?

[glass bane]

u and sd

[still sigil]

Give me numbers

for the mean and the variance of X

[glass bane]

mean is 54

v is 40.8

sd is 6.387

[still sigil]

Ok for the mean

[glass bane]

ye?

[still sigil]

Ok for the variance too

Well, great

Now we have something to work with

You can compute P(24 <= X <= 39) now right?

Since you know the distribution of X

[glass bane]

yes

[still sigil]

Good

Then I'll let you do that

[glass bane]

thank you so muchh 😭

[still sigil]

For question b, you can employ a similar strategy

Considering X = X1 - 2 X2 - 3 X3

[glass bane]

mhm yeah

[glass bane]

@still sigil sorry to disturb again but i just had a lil ques, like my answer is 0.00964 and the answer given is 0.0094 so does that matter a lot or no?

[still sigil]

Hey, no worries

Well there's a rounding error here so I would suppose it may matter

but then what method did you use?

[glass bane]

z score

[still sigil]

When did you round the standard deviation?

upon calculating it?

or at the end computation of the erf?

[glass bane]

like p(x-u/sd <z < 39-54/sd) one

so p(-4.697 <z< -2.34)

[still sigil]

did you round the standard deviation when you put it in the calculator?

[glass bane]

no

[still sigil]

or did you just straight up put the square root of 40.8

[glass bane]

i uh took up to 3 points

[still sigil]

try instead of rounding, put square root of 40.8

because the 40.8 is exact

[still sigil]

but this is approximate

[glass bane]

wait so still, what would be the z score for -4.697?

[glass bane]

one sec

[still sigil]

So basically, you compute:
erf((39 - 54)/sqrt(40.8)) - erf((24- 54)/sqrt(40.8))

in your calculator

[glass bane]

💀 i dont have that in my calc

😭

we are only allowed the basic scientific calc so

[still sigil]

Ok sorry

not erf

but you know

the cdf of the standard normal distribution

the z score

[glass bane]

ye?

[still sigil]

So if you have F the z-score function

F((39 - 54)/sqrt(40.8)) - F((24- 54)/sqrt(40.8))

[glass bane]

no 😭

[still sigil]

What do you even have on your calculator

sorry I'm asking because I dropped my calculator 4 years ago

[glass bane]

the most basic shit

[still sigil]

[glass bane]

why

[still sigil]

and what do you have for statistics

[still sigil]

because i do code and AI and have no use for one

[glass bane]

the literal most basic casio scientific calc

cos they wont allow us the advanced ones

[glass bane]

oh wow 😭

[still sigil]

then let me reformulate the question

[glass bane]

cant i just use the tables?

[still sigil]

how do you compute P(X <= t) for X a standard normally distributed variable

with a calculator

[glass bane]

we're taught with the tables so i have 0 idea

[still sigil]

Ok well... that sucks

do you have a picture of your table

[glass bane]

ye

one sec

[still sigil]

sure tyt

[glass bane]

is that even readable

[still sigil]

yeah that table doesn't expand very far

doesn't even cover the value we want P(Z <= -4.6967) so we're forced to round it to 0

[still sigil]

I would not blame you for not getting it right on this

I would blame your table and whoever came up with those bounds

[glass bane]

yeah

[glass bane]

😭 oh okay wow

[glass bane]

even on the net tho, i could only get it as far to 3.9?

can i ask you for help on the second part tho?

[still sigil]

I will answer in 15 but yes

[glass bane]

ah okay thankss

[unborn flare]

@glass bane has given 1 rep to @still sigil

[glass bane]

because i dont get that second part at all

damn 2am 💀

ima go sleep

[still sigil]

What is it that you do not get

[glass bane]

oh nvm, i got it