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

3 Asked on December 13, 2021

1 Asked on December 13, 2021 by rino

2 Asked on December 13, 2021 by mjoseph

1 Asked on December 13, 2021 by alexander-cska

analytic geometry coordinate systems linear algebra linear transformations matrices

1 Asked on December 13, 2021 by jon-g

1 Asked on December 13, 2021

3 Asked on December 13, 2021 by saganritual

1 Asked on December 10, 2021 by gcc

2 Asked on December 10, 2021 by yash-bhatia

2 Asked on December 10, 2021 by cheese12345

1 Asked on December 10, 2021 by paranoid

1 Asked on December 10, 2021 by sottoj

1 Asked on December 10, 2021 by billy-senders

2 Asked on December 10, 2021 by user6767509

1 Asked on December 10, 2021 by lce

absolute convergence convergence divergence infinite product

2 Asked on December 10, 2021

1 Asked on December 10, 2021 by mahesh87

1 Asked on December 10, 2021

0 Asked on December 10, 2021 by lightbulb

Get help from others!

Recent Questions

- 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?
- Does Google Analytics track 404 page responses as valid page views?

Recent Answers

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

© 2023 AnswerBun.com. All rights reserved. Sites we Love: PCI Database, UKBizDB, Menu Kuliner, Sharing RPP