Question

I'm learning discrete distributions and can't make heads or tails of it. How do I know when to plug in what? What am I missing?

Answer 1

It'd be easier for the rest of us to help you if you'd share a couple of specific problems. Your question is far too general as it now stands.

Answer 2

ok, first question: Is cumulative distribution function what happens when you add up the probability distribution function until a certain X?

Answer 3

A CDF for a random variable \(X\) is defined as \(F_X(k)=P(X\le k)\).

Using some properties of discrete probabilities, you can rewrite the above as
\[\begin{align*}F(x)&=P(X<k)+P(X=k)\\
&=P(X\le k-1)+P(X=k)\\
&=P(X<k-1)+P(X=k-1)+P(X=k)\\
&\vdots\\
&=P(X<k-m)+P(X<k-m+1)+\cdots+P(X=k-1)+P(X=k)\\
&={\large\sum_{i=0}^{m}}P(X=k-i)
\end{align*}\]
for appropriate \(m\).

So you're thinking along the right lines - the *cumulative* distribution function is the *cumulative* probability up to a point where \(X=k\).

Referring to an example, let's assume a binomial distribution, which has PDF
\[P(X=k)=\binom nk p^k(1-p)^{n-k}\]
where
\[\begin{cases}0\le p\le1\\n=0,1,2,...\\k=0,1,2,...,n\end{cases}\]
Some more concrete details: let \(p=0.25\), \(n=4\), and say we want the cumulative probability up to \(k=2\).

Well, there are two fairly easy ways of doing this: (1) derive the formula for the CDF and plug in \(k=2\), or (2) simply add the probabilities. Which method you use is up to you, though you'll find at times one manner of calculating \(F(k)\) is more convenient than another.

I haven't memorized the binomial CDF, and the first method isn't that work-intensive:
\[\begin{align*}F(2)&=P(X\le2)\\
&=P(X=0)+P(X=1)+P(X=2)\\
&=\binom40(0.25)^0(0.75)^4+\binom41(0.25)^1(0.75)^3+\binom42(0.25)^2(0.75)^2\\\\
&=\cdots
\end{align*}\]