(a) Where is the maximum of a normal distribution N(75,5) ? (b) Does f(x) touch the X-axis at μ±

(a) Where is the maximum of a normal distribution N(75,5) ?...

Key Concepts

Normal Distribution

A normal distribution is a continuous probability distribution characterized by its bell-shaped curve. It is defined by two parameters: the mean, which determines the center of the distribution, and the standard deviation, which describes the spread of the data around the mean. This distribution is widely used in statistics due to its natural occurrence in many phenomena as described by the central limit theorem.

Mean

The mean of a normal distribution is the central value around which the data are symmetrically distributed. It serves as the balancing point of the distribution and is also the location of the maximum of the probability density function. In practice, the mean is used as a measure of central tendency.

Mode

The mode of a probability distribution is the value at which the probability density function attains its maximum. For a normal distribution, the mean, median, and mode coincide, meaning the maximum of the distribution occurs at the mean.

Standard Deviation

The standard deviation is a measure of the dispersion or spread of a set of values around the mean in a probability distribution. In the context of the normal distribution, a larger standard deviation results in a wider, flatter bell curve, while a smaller standard deviation produces a narrower, taller curve.

Asymptotic Behavior

Asymptotic behavior refers to the manner in which the values of a function approach a given value as the input grows large in magnitude (positive or negative). In the case of the normal distribution, the probability density function approaches zero as x moves away from the mean, but it never actually reaches zero. This means the curve gets indefinitely close to the x-axis but never touches it, illustrating the concept of asymptotes.

Transcript

00:01 Okay, so first we know that f has a mean of zero and has a standard deviation of 1.

00:09 And i want first the probability that x will be less than, let's see, x then are equal to 1 .2.

00:19 So this is pretty straightforward stuff.

00:21 We know that we have this formula, which is if x is greater than b, then we will have f of b minus mu over standard deviation, which will just equal, in this case, 1 .2, right, whenever we plug this in.

00:42 So, f of 1 .2.

00:44 And our f equation is 1 over the square root of 2 pi, the integral from negative infinity to 1 .2, of e to the negative x squared over to dx, which of course we will not evaluate by hand.

01:03 With calculator.

01:05 As i've said in many past videos, if this is something you use in a calculator, and you're stumped at the negative infinity, you can just replace it with some arbitrary, large number.

01:15 So if you do that, what you'll find is that x will be very, very close to 0 .00, i'm sorry, 0 .885, right? so now b, b is a little bit different, because we need to know what will happen whenever, let's see here, x is greater than or equal to negative point four.

01:41 Okay.

01:42 So that's going to be handled a little bit differently.

01:45 One way that we could think of it is i could think of, well, yeah, let me do that again.

01:53 So i could say that if i want some, let's say, some point b, right, i want the area of all of this stuff right here.

02:03 So i could either, one of the easiest ways to do that is take the area of the entire curve and then subtract the area of everything up to that point.

02:16 So that's kind of what we're going to do here, because this is a probability density function.

02:22 I know the area under the entire curve is one, so then i'm going to subtract the area up to this point, right? so that basically means i'm going to subtract f of negative point four minus my mean over my standard deviation.

02:41 So you're just going to get 1 minus 4 of negative point 4.

02:46 So you're going to get 1 minus 1 over square root of 2 pi of the integral from negative infinity to negative 0 .4...

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