Mathematics Asked by PortMadeleineCrumpet on January 5, 2022

For questions dealing with insurance problems, I will often run into questions that involve finding aggregate variance. In a certain problem, I’m told that there are 50 policies, and various different amounts of claims with their own various probabilities for said claims. Once I’ve found the expected claim amount per 1 policy, the solution says that I should simply multiply the variance I’ve found by 50. Why is this the case, and not that if $S=50X$ then $Var[S]=Var[50X]=50^2Var[X]$?

The sum of independent r.v.'s is not $50$ times one of them. So the variance properties are that if $X_1,ldots,X_n$ are independent (or pairwise uncorrelated) then
$$
text{Var}(X_1+ldots+X_n)=text{Var}(X_1)+ldots+text{Var}(X_n).
$$
So for $50$ identically distributed and *independent* summands it will be $50text{Var}(X_1)$.

Answered by NCh on January 5, 2022

Multiplying by 50 the variance of one variabile or summing the same variance of 50 variabiles is not the same thing.

$V[50X]=50^2V[X]$

But if $X_i stackrel{d}=X_j$

(All the random variabiles have the same distribution and thus also the same variance, say $V[X]$)

$V[X_1+X_2+...+X_{50}]=50V[X]$

This, IF THE RV'S ARE UNCORRELATED. Otherwise you must sum all the covariances.

$$V[sum_i X_i]=sum_i sum_j Cov(X_i,X_j)$$

These results are very easy to be proved and you can find the proofs in any elementary statistic textbook

Answered by tommik on January 5, 2022

Get help from others!

Recent Answers

- Lex on Does Google Analytics track 404 page responses as valid page views?
- Peter Machado on Why fry rice before boiling?
- Jon Church on Why fry rice before boiling?
- Joshua Engel on Why fry rice before boiling?
- haakon.io on Why fry rice before boiling?

Recent Questions

- How can I transform graph image into a tikzpicture LaTeX code?
- How Do I Get The Ifruit App Off Of Gta 5 / Grand Theft Auto 5
- Iv’e designed a space elevator using a series of lasers. do you know anybody i could submit the designs too that could manufacture the concept and put it to use
- Need help finding a book. Female OP protagonist, magic
- Why is the WWF pending games (“Your turn”) area replaced w/ a column of “Bonus & Reward”gift boxes?

© 2024 TransWikia.com. All rights reserved. Sites we Love: PCI Database, UKBizDB, Menu Kuliner, Sharing RPP