Question

minerals produced from mines 1, 2, 3 and are independent normal random variables with means 80, 90, 75 pounds, respectively. what is the probability that the combined amount of mineral produced from all three mines exceeds 283 pounds? [Hint: use sum random variable Y = X1 + X2 + X3

Answer 1

0%. Even if you get 3 of the heaviest mineral, it only gives you 270 lbs (unless you means something else)

Answer 2

answer is .0351
you may need to use linear combinations

Answer 3

What would be the constant? To you take out 1 mineral/mine or what?

Answer 4

I think some info is missing.

Answer 5

no other information is given sorry

Answer 6

standard deviations 12, 14, 10
forgot crucial information!

Answer 7

I got .036 do you know about moment generating functions?

Answer 8

some what
the e^tx thing?

Answer 9

yes it's the expected value of e^tx. Do you know the moment generating function for a normal random variable?

Answer 10

no but i can look it up

Answer 11

i can type it for you...

\[M(t)=e^{\mu*t+\sigma^{2}t^{2}/2}\]

Answer 12

there's so many variations of formulas i can't retain them in my memory

Answer 13

that has mu as the mean and sigma as the standard deviation

Answer 14

i've seen that before but didn't actually had to solve any questions with it so far

Answer 15

do i use that formula?

Answer 16

you use it to derive something that is helpful to remember about sum of independent normal vars. Do you want me to derive the result or just tell you what it is?

Answer 17

you can just tell me what it is lol

Answer 18

ok haha. The mean of the sum of independent normal vars is the sum of the means, and the variance is the sum of the variances. So we have mean 80+90+75=245 and variance = 144+196+100=440

Answer 19

then you convert it to standard normal and find the probability of being greater than 283 by looking a a standard normal chart

Answer 20

do you know how to convert it?

Answer 21

what is standard normal? and i don't think we can use standard normal chart for this

Answer 22

standard normal is normal with mean 0 variance 1. Well if you can't use the chart you just integrate. Do you know the pdf for normal variables?

Answer 23

yeah

Answer 24

ok so you have the variance and mean, so you know the pdf and you integrate from 283 to infinity

Answer 25

i use the standard pdf formula?

Answer 26

yes you can use that. It's kind of a mess though I usually use standard normal

Answer 27

im looking at the integral now its pretty bad. I think I'd have to do it numerically. You haven't used standard normal in your class before?

Answer 28

hmm ok i'll have to play around with it for a while

Answer 29

no we've only used charts for z-score

Answer 30

everything else we just integrated or didn't use charts

Answer 31

yeah z score, thats standard normal

Answer 32

oh haha

Answer 33

you have mean and standard deviation so use the z score to find probability

Answer 34

i only knew it as table VI on pg 382 lol

Answer 35

haha

Answer 36

using the z = (x-m)/st.dev?

Answer 37

yeah that's it. stddev is sqrt( sum of the variances of each variable) mean is sum of the means

Answer 38

oh ok thanks. are you an actuary?

Answer 39

im a graduate student in math studying modern algebra. I just took a grad probability class last semester though.

Answer 40

ahh nice

Answer 41

are you in college?

Answer 42

no i graduated recently but taking some courses at local college

Answer 43

need it for grad school

Answer 44

are you taking a prob/stats class now? i noticed a bunch of your questions are from that topic

Answer 45

probability and stat is worse than multivariable calc for me

Answer 46

yeah

Answer 47

i like prob not stats. Did my answers to your other questions show up? Im not sure it worked.

Answer 48

yeah they did thanks. but haven't worked on them

Answer 49

but i have a good idea on how to solve them now though

Answer 50

good. well ill cya around the site I just found it and I think its fun. good luck with your probability.

Answer 51

thanks!!