I m not understanding CDF

[semidevil]

I"m not understanding CDF...

My textbook doesn't seem to give enough information on it.

I"m not understanding what do to when involved with CDFs in general.

So if I want to convert a pdf to cdf, given a function
fy(Y) =

0 for y <0
2y for 0 <= y <= 1/2
6 - 6y for 1/2 < y <= 1
0 for y > 1

so the book does this:
for 0 <= y <= 1/2, it integrates (0 dt from -infinite to 0) + the integeral of (2dt from 0 to y).

answer is y^2.

then, for 1/2 < y <= 1, it does the integeral of 0dt from -infinite to 0,+ the integeral of 2dt from 0 to 1/2dt, + the integeral of 6 - 6t from 1/2 to y.

answer is 6y - 3y^2 - 2.

and y > 1, = 1.

so we get the new cdf.

the book doesn't give any explanation of what it did. So what just happened here? how did they do all that just by looking at the function? why did they integrate from - infinite of 0, and then from 0 to y, and then 0 to 1/2.

well, you get my point...I don't know how to approach those problems..

 

[matt grime]

Given a PDF, f, the associated CDF is usually F(k):= P(X<k), which is exactly the integral from -inf to k of f. They split F into several parts in this example because the PDF is defined in several parts.

 

[tacman]

First of all, I can understand your frustration with not being able to fully grasp the concept of CDF (Cumulative Distribution Function). It can be a complex topic to understand, especially if your textbook doesn't provide enough explanation or examples.

Let me try to break it down for you. CDF is a function that represents the probability that a random variable will take on a value less than or equal to a specific value. In simpler terms, it tells you the likelihood of a certain event occurring within a given range.

In your example, you are given a probability density function (pdf) fy(Y) which gives the probability of a random variable Y taking on a certain value. To convert this pdf to a CDF, you need to integrate the pdf from negative infinity to the point you are interested in. In your case, you are interested in finding the CDF for values 0 <= y <= 1/2, 1/2 < y <= 1, and y > 1.

For 0 <= y <= 1/2, the CDF is the integral of the pdf from negative infinity to 0, which is 0, plus the integral from 0 to y. This is because the probability of Y taking on values less than 0 is 0, and the probability of Y taking on values between 0 and y is given by the integral of the pdf from 0 to y.

Similarly, for 1/2 < y <= 1, the CDF is the integral of the pdf from negative infinity to 0, which is 0, plus the integral from 0 to 1/2, plus the integral from 1/2 to y. This is because the probability of Y taking on values less than 0 is 0, the probability of Y taking on values between 0 and 1/2 is given by the integral of the pdf from 0 to 1/2, and the probability of Y taking on values between 1/2 and y is given by the integral of the pdf from 1/2 to y.

Finally, for y > 1, the CDF is simply 1, as the probability of Y taking on values greater than 1 is 0.

To summarize, the CDF is calculated by integrating the pdf from negative infinity to the point of interest, and adding the integrals of any ranges that fall within that point.