How can I calculate the PMF from a given CDF?

[EugP]

How do I find PMF if I have CDF? For example:

[tex]
F_X (x)=\left\{\begin{array}{cc}0,&\mbox{ if }x\leq -1
\\0.2, & \mbox{ if } -1\leq x < 0
\\0.7, & \mbox{ if } 0\leq x < 1
\\1, & \mbox{ if } x \geq 1
\end{array}\right.[/tex]

 

[EnumaElish]

f(x) = [itex]\Delta F(x)/\Delta x[/itex]

 

[tacman]

To find the PMF (Probability Mass Function) from the given CDF (Cumulative Distribution Function), you can use the following formula:

P(X=x) = F_X(x) - F_X(x-1)

Where P(X=x) is the probability of the random variable X taking the value of x, F_X(x) is the CDF at x, and F_X(x-1) is the CDF at x-1.

Using this formula, we can calculate the PMF for the given example. Let's take x = 0 as an example:

P(X=0) = F_X(0) - F_X(-1)

= 0.7 - 0.2

= 0.5

Similarly, for x = 1:

P(X=1) = F_X(1) - F_X(0)

= 1 - 0.7

= 0.3

And for x = -2:

P(X=-2) = F_X(-2) - F_X(-3)

= 0 - 0

= 0

In this way, we can find the PMF for any value of x within the given range. The PMF is useful in determining the probability of discrete random variables, while the CDF is useful in determining the probability of a range of values for a continuous random variable.