(STATISTICS) 3 randomly selected observations from standard norm dist

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Homework Help Overview

The problem involves selecting three random observations from a standard normal distribution and determining the probability that their sum is less than 2.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the interpretation of the problem, with some attempting to relate the sum of the observations to their average. Questions arise regarding how to add normal distributions and the distribution of the sum itself.

Discussion Status

Some participants have provided insights into the nature of the sum of normally distributed variables and the need to find its probability density function. There is an ongoing exploration of the concepts involved, but no consensus has been reached regarding the method to solve the problem.

Contextual Notes

One participant mentions a specific numerical answer but expresses uncertainty about the method to arrive at it. There is a focus on understanding the underlying statistical principles rather than providing a direct solution.

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Homework Statement



3 randomly selected observations form the standard normal distribution are selected. What is the probability that their sum is less than 2?

Homework Equations





The Attempt at a Solution



I know that the answer is 0.874928, but I don't know how to get that.


In my mind, you would say that 2/3=0.67, so what is the probability that the average of the 3 values is less than 0.67?

This gets me the wrong answer however.
 
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How do you add normal distributions?
 
I think what the question is asking, is that 3 random numbers are chosen FROM a standard normal distribution. It's then asking what the chances are that their sum is less than 2.
 
Same thing.
The numbers are x,y, and z - their possible values are distributed normally with mean 0 and standard deviation 1.

the sum of them is s=x+y+z ... s has a range of possible values too.
You need to find how s is distributed, it's pdf, so you can find P(s<2). How do you add normal distributions?
 

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