Provide two interpretations of the area under the graph of a probability density function.

Name: Problem 3, Chapter 7: Fundamentals of Statistics
Uploaded: 2021-06-22T19:22:49-08:00
Duration: 5 min 7 s
Channel: Barsha Rana
Description: Provide two interpretations of the area under the graph of a probability density function.

Provide two interpretations of the area under the graph of a...

Key Concepts

Relative Frequency Interpretation

Interpreting the area under the PDF as a relative frequency means that if the experiment were repeated an infinite number of times, the area over a specific interval would approximate the long-run proportion of observations or outcomes that fall within that interval. This view aligns the geometric area with empirical frequencies, ensuring consistency between theoretical probability and observed data trends.

Probability Interpretation

The area under the curve of a probability density function (PDF) between two points represents the probability that a continuous random variable falls within that interval. Since the entire area under the PDF is one, any partial area corresponds to a valid probability measure, quantifying the likelihood of the event occurring in that range.

Transcript

00:01 Welcome to numerate.

00:02 In the current problem, we are going to discuss two interpretations for the under curve, okay? and as a problem -reensity function.

00:22 So when we are considered in a pdf, first thing we remember is it is a continuous variable.

00:33 Now, a continuous variable will be having a pdf which can be a of any shape, but what we have learned so far is something like this.

00:51 And i know you know the name already.

00:53 It is the pdf of a normal distribution.

00:57 Now, what does it mean? what is the two interpretation? first thing is like for normal distribution, the entire range of x is the real line.

01:09 Okay? so over the entire range, the total area under it should be that is if it is the lowest bound a and if it is the highest bound b and suppose this function is called g x then if you integrate g x from the lowest bound to our highest bound it should always be one this is one one interpretation the second thing is this is always positive we have seen many kind of mathematical functions like this okay the sign function the cost function the tan function right but those all have some negative values also but the pdf has to be zero for all values of x okay now another assumption interpretation would be individual value if you ask me say this particular point say it is c okay so suppose a and this b includes c so at exactly that point the pdf the the the the the probability you might be thinking okay this is some height so it will be some say this five no it should be zero because there are infinite number of points, correct? so, for each point if i consider the probability, what will happen? we just told a to b, gx, d, x, should be one.

02:57 Then this assumption will get violated.

02:59 How? integration a to b is nothing but a giant sum that starts from a and finishes at b...

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