Normal random variables

In summary, the proof shows that the function \psi(x) = 2\phi(x) - 1, also known as the positive normal distribution, can be derived from the standard normal distribution by taking the absolute value of Z. This is shown by starting with the cumulative distribution function of |Z| and simplifying it using the property that \phi(-x) = 1 - \phi(x).
  • #1
Proggy99
51
0

Homework Statement


Let [tex]\psi[/tex](x) = 2[tex]\phi[/tex](x) - 1. The function [tex]\psi[/tex] is called the positive normal distribution. Prove that if Z is standard normal, then |Z| is positive normal.


Homework Equations





The Attempt at a Solution


I am not really sure where to begin with this. Can anyone provide me a jumping off point, please?

I do know that [tex]\phi[/tex](-x) = 1 - [tex]\phi[/tex](x)
and so [tex]\phi[/tex](x) + [tex]\phi[/tex](-x) - 1 = 0.
I am not sure how to utilize that or if it is even on the right track. Thanks for any help.
 
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  • #2
Start with the cumulative distribution function of [tex] |Z| [/tex]

[tex]
P(|Z| \le x) = P(-x \le Z \le x) = \Phi(x) - \Phi(-x)
[/tex]

where [tex] \Phi [/tex] is the cdf of the standard normal. How can you simplify [tex] \Phi(-x) [/tex]?
 
  • #3
Ahhh, that makes perfect sense statdad. I kept trying to factor out 'x' when I looked at it the way you did it and got nowhere so discarded that method. I would substitute that with the equation I put in my first post to get the equation from the definition of positive normal. I just could not come up with that middle step to link the ideas until you put it so plainly. Thanks!


statdad said:
Start with the cumulative distribution function of [tex] |Z| [/tex]

[tex]
P(|Z| \le x) = P(-x \le Z \le x) = \Phi(x) - \Phi(-x)
[/tex]

where [tex] \Phi [/tex] is the cdf of the standard normal. How can you simplify [tex] \Phi(-x) [/tex]?
 

1. What is a normal random variable?

A normal random variable is a type of continuous random variable that follows a bell-shaped distribution known as the normal distribution. It is often used in statistics and probability to model real-world phenomena.

2. How is a normal random variable different from other types of random variables?

A normal random variable differs from other types of random variables in that it follows a specific mathematical distribution, the normal distribution. This distribution is characterized by its bell-shaped curve and has certain properties, such as the mean, median, and mode all being equal.

3. What is the central limit theorem and how does it relate to normal random variables?

The central limit theorem states that as the sample size increases, the distribution of sample means will approach a normal distribution, regardless of the underlying distribution of the population. This means that many real-world phenomena can be approximated by a normal random variable, making it a useful tool in statistical analysis.

4. How do you calculate probabilities for a normal random variable?

To calculate probabilities for a normal random variable, you can use the standard normal distribution table or a statistical calculator. These tools will give you the probability of a certain value falling within a given range on the normal distribution curve.

5. What types of real-world phenomena can be modeled by a normal random variable?

Many real-world phenomena can be modeled by a normal random variable, including heights, weights, test scores, and IQ scores. This is because these types of data tend to follow a bell-shaped distribution in large populations. Additionally, many statistical tests and analyses assume a normal distribution, making the normal random variable a valuable tool in data analysis and research.

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