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## Statistics and probability

### Course: Statistics and probability>Unit 4

Lesson 7: More on normal distributions

# Deep definition of the normal distribution

We take an extremely deep dive into the normal distribution to explore the parent function that generates normal distributions, and how to modify parameters in the function to produce a normal distribution with any given mean and standard deviation. We also look at relative frequency as area under the normal distribution. Created by Sal Khan.

## Want to join the conversation?

• Why does the formula for a normal distribution contain pi? •   For normalization purposes. The integral of the rest of the function is square root of 2xpi. So it must be normalized (integral of negative to positive infinity must be equal to 1 in order to define a probability density distribution). Actually, the normal distribution is based on the function exp(-x²/2). If you try to graph that, you'll see it looks already like the bell shape of the normal function. If you then graph exp(-(x-mu)²/2), you'll see the same function shifted by its mean - the mean must correspond to the function's maximum. That is a basic characteristic of the normal distribution. Finally, if you try out exp(-((x-mu)/sigma)²/2) you'll then find out that you have the same shape shifted by the mean and elongated or shrunk by the standard deviation. The same applies to any function f(x). f(x/2) is tighter, while f(x/0.5) is wider than the original f(x). So the core of the normal distribution is exp(-x²/2). The variable squared gives this function is parabolic look, while the negative sign makes its concavity look downward. At last, the exponential gives the function its asymptotic behavior. For more information on the nature of the normal distribution, take a look at http://courses.ncssm.edu/math/Talks/PDFS/normal.pdf
• After listening to this video, and reflecting on what I learned, I came across this thought: "The probability of something occuring 0% of the time" is not an equivelant statement to "It cannot happen". If my thought process is right, this is because 0% is on the chart. However, "cannot happen" is a statement that is not on the chart.

So, the probability of randomly pulling data ten-thousand standard deviations away might be 0%, but it is still on the normal distribution curve. The probablity of nighttime and daytime occuring simotaniously cannot happen. So that is not on the curve.

Thus, '0% chance of happening' is not an equivelant statement to 'cannot happen'. Right? •   This is a good question about probability. The shortest answer would be: having a probability of zero is equivalent with being impossible. In fact, that is how we define impossible. The rest of this answer is a somewhat lengthy explanation, but I couldn't think of a way to shorten it without sacrificing a point.

What probability does is assign a value (the probability) to a particular outcome. Say I place 3 red marbles, 3 yellow marbles. How can we describe the probability distribution? Well, we'd start with the obvious:

P( red )=3/6
P( yellow ) = 3/6

But we can always write things like:

P( bananna ) = 0
P( Battlestar Galactica ) = 0
P( coffee mug ) = 0
... etc.

We're simply only interested in the non-zero probability, so we don't write all the items for which the probability is zero.

Now we get to the normal distribution. You are right that on a theoretical level, it goes out to infinity in either direction. But the curve never actually hits zero. So all of those values are possible values, they just have extremely small probability. What you are thinking about as 0% probability, is actually just rounded off. If something has probability 0.0001, okay, that's pretty unlikely. What about 0.0000001 ? Well, that's even more unlikely. At some point, we just get sick of the whole process and round it off to 0.

Are those values actually impossible to obtain? Possibly, but not necessarily. However, in spite of whether or not something is theoretically possible, on a practical level it is impossible. For example, let's say we're measuring the rainfall in a certain city, and that the mean is 25 inches and the standard deviation is 5 inches.

Clearly the amount of rainfall cannot be negative, but we can still put a normal distribution on this. Why? Because the probabilities down near zero will be so small that they'll round off to zero, so it makes no difference really. Let's try it out, say we want to calculate the probability of less than -1 inches of rain. Doing this, we'd get P(X<-1)=0.0000001.

It's clearly an impossible event, but the probability is not equal to zero. However, it is so small as to be practically zero, so this isn't going to affect much of anything, and it is much easier than the alternative of using a different distribution to model the rainfall.

Similarly, we could ask about the probability of more than 60 inches of rain. There's not really an upper bound on rainfall, so 60 inches isn't a physically impossible value like -1, but the probability is 0.000000000001. Again, this is so small that we'd just round it off to zero, and say that it is impossible to have more than 60 inches of rain in this area.

Sorry for the length of this, but it was an interesting question that is hard to describe in a shorter post.
Does anyone else have this problem and how can we solve it? • Yes, I had this problem and the parent directory also doesn't have a link to the file. Just google the link and you'll find it in the results (I found it on weebly something).

I also opened a bug report - "Issue 17659:Missing download from class (404)"
• My teacher has only taught me how to key in the numbers into the formula instead of teaching how the formula itself has been derived! Does anybody out there face the same situation as I do? • If we draw two vertical lines at each extrema of the standard deviation will they always cross the graph at the inflection points? (The point where the graph transitions from concave up to concave down). If this is the case, it would make more sense why a decrease in standard deviation causes the graph to become more narrow. • Is it possible to convert uniform distribution to normal distribution? I mean, that randomizer gives only uniform distribution and i need Gaussian.I heard about inverse transformation method for conversion from uniform to Cauchy distribution. Is there any recipes for making normal distribution, based on uniform? • If you're trying to create a distribution that is approximately Normally distributed (and you only have Uniformly distributed random variables), one possible approach--thanks to the Central Limit theorem--is to take n uniform r.v.'s (for instance, n=10), calculate their mean, and then repeat a bunch of times. Then, if your original r.v.'s were Uniform with mean=μ and variance=σ², your distribution of sample means will be (approximately) N~(μ, σ²/n).
• Sal talks, towards the start of this video, about integral calculus being very helpful for this. If I'm going through all the maths on khan academy, is it worth me leaving this for now, doing the calculus modules, and then coming back to this?    