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Describe the standard normal distribution. What are its characteristics?

$$\int_{a}^{b} \frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{x^{2}}{2}} d x$$

Calculus 2 / BC

Chapter 11

Probability and Calculus

Section 3

Special Probability Density Functions

Taylor Series

Harvey Mudd College

Baylor University

Idaho State University

Lectures

01:38

In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point. The Taylor series of a function "f" is the limit of that series, evaluated at a particular point "x".

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What are the characteristi…

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List the major characteris…

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What is the mean of the st…

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What is the difference bet…

in number 15 were asked, What is the standard distribution curve? What are its characteristic? I think this would have been a great number one, because I've already been using the normal distribution. But whatever, that's answer it now what can we tell about it? There is no answer in the manual for this, so you'll have to on bear with me and consider that there might be more to it than what I found. But here, the character six I found, I think are really important to the standard normal, the standard normal distribution. So this is not any standard with normal distribution. This is the bell shaped one that has, I mean of zero and a sigma of one signal being there. Standard deviation. So those are two very important characteristics of the curve. Another one week tell is that the total area is one. I mean, this is always the case, but it's for any probability density function. But it's also a characteristic, this specific one, which is an important one. Um, we could say it goes from minus infinity two plus infinity, um, which maybe I should write differently. Instead, we should say that X is between minus and Finney and plus in Philly. If we see this accesses X here. So this job goes all the way to my cousin Fannie and all the way to a pleasant day. Um and then and everything we say about it is describe it as a function. Um, so here you can look in your manual and see that you got something like that here. Excellent, too. So this is a simplification off. The normal aren't of the more general bell shaped curve because the means zero and this tender division is one. That's the simplest form of it that we can get. Some of you might ask why is there this factor here? Why don't we just remove that if we want to? Simpler form? Well, the answer to that is that the total area has to be one. If we remove this factor, the total area wouldn't be one anywhere. It would be more than one and has to be one, because total probability that any event happens is 100% can't be 200% because that doesn't make any sense. Right? So that's the role that this factor is playing here. Um, that's it for me. I think that's those are all the most important characteristics You could probably find more, but I'm not sure you could find a lot more. That's that's really whether is to discourage, all right?

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