(STATISTICS) 3 randomly selected observations from standard norm dist

[skyturnred]

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.

 

[Simon Bridge]

How do you add normal distributions?

 

[skyturnred]

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.

 

[Simon Bridge]

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?