Probability Distribution Function

[p.mather]

Homework Statement

Given the probability distribution function:

See attachment.

Determine the:

1. Probability density function.
2. The mean.
3. The median.

Homework Equations

Hello,

I am really struggling with this subject area, this is an example I have found, would someone be able to go through a solution so I can begin to understand it a bit more.

Appreciate any help.

Thanks.

The Attempt at a Solution

 

[LCKurtz]

It's pretty hard to work with that "example" since it is neither a probability density function nor a cumulative distribution function.

Question for you: How do I know that?

 

[Stephen Tashi]

The probability density function is F'(x) , the derivative of F(x).
The mean is $$\int x F'(x) dx$$ taken over the interval where F'(x) is not zero.
The median is found by solving for x in the equation F(x) = 1/2

(The mode (or modes) is found by finding the value (or values) where F'(x) is maximum. )

I don't have time to go through this in detail now. If you have further questions, ask.

 

[p.mather]

The probability density function is F'(x) , the derivative of F(x).
The mean is $$\int x F'(x) dx$$ taken over the interval where F'(x) is not zero.
The median is found by solving for x in the equation F(x) = 1/2

(The mode (or modes) is found by finding the value (or values) where F'(x) is maximum. )

I don't have time to go through this in detail now. If you have further questions, ask.

As i was saying i am not good at this at all and its something i need to begin to understand. Would you please be able to provide a worked solution so i can begin to understand. It would be greatly appreciated. Thanks.

 

[Pyrrhus]

It's pretty hard to work with that "example" since it is neither a probability density function nor a cumulative distribution function.

Question for you: How do I know that?

Do not mindlessly start taking derivatives of that function. First Re-read LCKurtz post.

What is the range of a CDF?, and what are the restriction for the range of a PDF?

What are the requirements for a function to be a CDF or a PDF?

 

[Stephen Tashi]

The cumulative distribution (sometimes simply called "the distribution") of a random variable $X$ is the function $F(x)$ that gives the probability that $X \leq x$.

The example you gave:

$$F(x) = 1 - e^{2x}$$ for $$x \geq 0$$ doesn't make sense as a cumulative distribution because for positive values of $x$ , $F(x) < 0$ and probabilities must be non-negative numbers.

One way to fix the typo in the example is say that $F(x)$ is defined by:

$$F(x) = 1 - e^{-2x}$$ for $x \geq 0$$$F(x) = 0$$ for $x < 0$

Is this much clear?