(STATISTICS) 3 randomly selected observations from standard norm dist

In summary, the conversation discusses the probability of the sum of three randomly selected observations from a standard normal distribution being less than 2. The correct answer is 0.874928, but the process to get to this answer is unclear. There is a question about how to add normal distributions and the possibility of finding the probability of the average of the three values being less than 0.67. The conversation also clarifies that the numbers are distributed normally with a mean of 0 and standard deviation of 1, and the final question is about finding the probability that the sum of the three numbers is less than 2.
  • #1
skyturnred
118
0

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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  • #2
How do you add normal distributions?
 
  • #3
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.
 
  • #4
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?
 

FAQ: (STATISTICS) 3 randomly selected observations from standard norm dist

1. What is a standard normal distribution?

A standard normal distribution is a probability distribution that follows a bell-shaped curve and has a mean of 0 and a standard deviation of 1. It is often used in statistics to model natural phenomena and is also known as the Gaussian distribution or the bell curve.

2. How are observations randomly selected from a standard normal distribution?

Observations from a standard normal distribution are randomly selected by using a random number generator. This generator generates numbers that are uniformly distributed between 0 and 1. These numbers are then transformed using a mathematical formula to follow a standard normal distribution.

3. Why are 3 observations chosen from a standard normal distribution?

Three observations are a commonly chosen number because it provides enough data points to make meaningful conclusions but is also manageable for analysis. Additionally, the Central Limit Theorem states that as the sample size increases, the sampling distribution of the mean approaches a normal distribution, so three observations can provide a good approximation of the standard normal distribution.

4. What is the purpose of selecting 3 observations from a standard normal distribution?

The purpose of selecting 3 observations from a standard normal distribution is to understand the characteristics and properties of the distribution. This can help in making predictions, drawing conclusions, and performing statistical tests. It also allows for the comparison of different distributions and to determine if a dataset follows a normal distribution.

5. Are the 3 randomly selected observations from a standard normal distribution always the same?

No, the 3 randomly selected observations from a standard normal distribution will not always be the same. This is because each time the observations are randomly selected, a different set of numbers will be generated from the random number generator. However, as the sample size increases, the sampling distribution of the mean will become more consistent and approach the standard normal distribution.

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