Normal Distribution: Mean & Std Dev for Conditional Expected Values

In summary, a normal distribution can be defined by two parameters - the mean and the standard deviation. It is possible to use these parameters to obtain conditional expected values for a normal distribution, by standardizing the distribution and using the properties of the standard normal distribution. However, the process becomes more complex when the mean and standard deviation are not equal to 1.
  • #1
Einstein
13
1
A normal distribution can be completely defined by two parameters - the mean and the standard deviation. Given a normal distribution however, say X, how can I use just the mean and the standard deviation to give me conditional expected values for X<=0 and for X>0? I am guessing the distribution can be standardised to obtain a z-statistic.
 
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  • #2
"Given a normal distribution however, say X" - I assume you mean that the variable [tex] X [/tex] has a normal distribution. Are both [tex] \mu [/tex] and [tex] \sigma [/tex] known?

"how can I use just the mean and the standard deviation to give me conditional expected values for [tex] X \le 0 [/tex] and for [tex] X>0 [/tex] ?"
This doesn't make sense to me as it stands. In statistics we take expected values of some function of a random variable - can you elaborate on what it is you seek?
 
  • #3
If X has normal distribution with mean [itex]\mu[/itex] and standard deviation [itex]\sigma[/itex] then [itex]z= (x- \mu)/\sigma[/itex] has the standard normal distribution. As statdad said, "conditional expected values for X< 0 and X> 0" makes no sense." I might interpret as "suppose X a standard normal distribution, restricted to be larger than 0. What is the the expected value of X?"

That would be
[tex]\frac{1}{\sqrt{\pi}}\int_0^\infty x e^{-x^2}dx= \frac{1}{2\sqrt{\pi}}[/tex]
The general problem, with non-zero mean or standard deviation not 1 would be a much harder integral.
 
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1. What is a normal distribution?

A normal distribution is a commonly observed pattern in many natural phenomena, where the data is symmetrically distributed around a central value. It is often referred to as a bell curve due to its shape. Most of the data falls within one standard deviation of the mean, with a smaller percentage falling within two or three standard deviations.

2. What is the mean in a normal distribution?

The mean, also known as the average, represents the central value of a normal distribution. It is calculated by summing all the data points and dividing by the total number of data points. In a normal distribution, the mean is also the peak of the bell curve.

3. What is the standard deviation in a normal distribution?

The standard deviation is a measure of how spread out the data is in a normal distribution. It is calculated by finding the square root of the sum of squared differences between each data point and the mean, divided by the total number of data points. It tells us how far the data points are from the mean, with a smaller standard deviation indicating a more tightly clustered distribution.

4. What is the relationship between the mean and standard deviation in a normal distribution?

The mean and standard deviation are both important measures in a normal distribution. The mean represents the central value, while the standard deviation tells us about the spread of the data. Together, they can be used to describe and analyze the data in a normal distribution.

5. How are conditional expected values calculated in a normal distribution?

Conditional expected values in a normal distribution are calculated by taking the mean and standard deviation of a subset of data that meets a specific condition. For example, if we want to find the conditional expected value for data points that fall within one standard deviation of the mean, we would use the mean and standard deviation of that specific subset of data.

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