# CDF vs. PDF

#### HikariT

##### New Member
Hey everyone,

I was looking for example "cdf" problems to do to get ready for a test and I came up with one that I didn't understand....
The following curve represents the cumulative density function for a random variable X.

[The picture is attached as a document]

Which of the following statements would be aspects of the original probability function of X?

I. The curve is symmetric

II. The curve is a uniform distribution

III. The median is 5.

I only
III only
I and III only
II and III only
I, II, and III

Can anyone explain? I'm freaking out cause I'm not really good at these.
Thanks.

#### mp83

##### TS Contributor
A CDF is definetely uniformly distributed,so II is in, and symmetry looks good

#### HikariT

##### New Member
Cool, that's what I guessed So if CDF is uniformly distributed/symmetric that means the orignal is too? And if the CDF's median is 5 does that mean the orignals median is 5 also? Thanks a lot.

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#### mp83

##### TS Contributor
Well, the median of a CDF ia always .5 as it splits the probability in half (.5 to the left-.5 to the right). Not the same for a PDF,though

#### HikariT

##### New Member
Ok, I understand that. If that's so though, wouldn't the answer have to be that the original's curve is symmetric / uniformly distributed? That'd be "I and II" only, but that wasn't one of the answer choices given of the site Is there a typo or something?

#### Mean Joe

##### TS Contributor
Which of the following statements would be aspects of the original probability function of X?

I. The curve is symmetric

II. The curve is a uniform distribution

III. The median is 5.

I only
III only
I and III only
II and III only
I, II, and III

Can anyone explain? I'm freaking out cause I'm not really good at these.
Thanks.
The pdf can be drawn from the attached cdf, using the property that the pdf is the derivative of the cdf. Think of the slope of the cdf.

Since the slope of the attached cdf is the same for the left portion as the right portion, then the pdf is symmetric.

If the pdf were uniform, then it would be flat; thus the cdf would have to have a constant slope throughout (a line). From the attached cdf (starts out with a small slope, then increases in slope up to the median, then decreases in slope), I would guess that the pdf is bell-shaped.

You can find the median directly from the graph of the cdf--look for the x-coordinate of the point on the cdf where the y-coordinate = 50%.

#### mp83

##### TS Contributor
Ok,so looking at the graph I answered as if the question was about the CDF. The symmetry still holds, (bell-shaped probably as Mean Joe said).The rest not.

#### TheEcologist

##### Global Moderator
The pdf can be drawn from the attached cdf, using the property that the pdf is the derivative of the cdf. Think of the slope of the cdf.

You can find the median directly from the graph of the cdf--look for the x-coordinate of the point on the cdf where the y-coordinate = 50%.
I dont think you can put it anymore clearly without giving the aswer away.

Is there a typo or something?
There is no typo the correct answer is in the list.