P(x=mean) of normal PDF with low sigma - not allowed?

In summary, the conversation discusses the normal probability distribution and its formula, which can result in an absurdity when σ<(1/sqrt(2π)). It also brings up the question of how a change in scale on the horizontal axis can affect the probability of the mean. The experts explain that for any continuous PDF, the probability of x being equal to a specific value is 0, and what can be calculated is the probability of x being less than or equal to the mean.
  • #1
nomadreid
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P(x=mean) of normal PDF with low sigma -- not allowed?

About the normal probability distribution: with formula
P(X=x) = (1/(σ*sqrt(2π))*exp(-(x-μ)2/2σ2), what happens when you look at P(X=μ) if σ<(1/sqrt(2π))? You get P(X=μ)>1, an absurdity. What is going on?

Second question is one about intuition: suppose μ=0, then why would a scale change of the horizontal axis (say by changing units from meters to kilometers) , which would also change σ, affect the probability of the mean, which it would by the formula?
 
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  • #2


the probability is 0 since what you looked at is a probability density p(X in x,x+dx) equals f(x)dx, where f is your gaussian function
 
  • #3


For any continuous PDF, the probability that x is equal to any specific value, rather than in a given interval or set, is 0.
 
  • #4


Thanks, jk22 and HallsofIvy. My reaction upon reading your replies and thinking for a couple of seconds was, "Of course. Stupid of me." So thanks for that destruction of the mental block.
 
  • #5


What you can calculate is
[tex]P(X \le \mu) = \int_{-\infty}^{\mu} p(x) \, dx[/tex]
where p(x) the PDF. Of course, this will give you 1/2 independent of the mean or standard deviation.
 

FAQ: P(x=mean) of normal PDF with low sigma - not allowed?

1. What is a normal PDF with low sigma?

A normal PDF (probability density function) with low sigma refers to a normal distribution with a small standard deviation. This means that the data points are clustered closely around the mean, resulting in a tall and narrow bell curve shape.

2. What does it mean for P(x=mean) to not be allowed?

In the context of a normal PDF with low sigma, P(x=mean) refers to the probability of a data point having the exact same value as the mean. This is not allowed because the probability of any specific data point having an exact value in a continuous distribution is always zero.

3. How does a low sigma affect the shape of a normal distribution?

A low sigma, or small standard deviation, results in a tall and narrow bell curve shape for a normal distribution. This means that the data points are clustered closely around the mean, resulting in a smaller spread of values compared to a normal distribution with a higher sigma.

4. Can a normal PDF with a low sigma have a negative mean?

Yes, a normal PDF with a low sigma can have a negative mean. The mean of a normal distribution represents the central tendency of the data, and it can be positive, negative, or zero depending on the data set.

5. How is the area under the curve affected by a normal PDF with a low sigma?

A normal PDF with a low sigma will have a smaller area under the curve compared to a normal distribution with a higher sigma. This is because the data points are clustered closely around the mean, resulting in a smaller spread of values and a narrower curve.

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