Can you explain to me discrete distributions?

1. Random Variable

A random variable $X$ is a rule that gives you a number after an experiment.
If the experiment outputs only separate, countable numbers (0,1,2,…), then $X$ is discrete.

2. Probability Mass Function (PMF)

For a discrete variable we write a table or formula called the probability mass function (PMF):

The two big rules are:

1. $0 \le p(x) \le 1$ for every $x$
2. $\sum_{x} p(x) = 1$ (the total chance must be 1)

3. Common Discrete Distributions

4. Mean and Variance

For any discrete distribution:

Mean (expected value):

Variance:

5. Working Steps

1. Identify what counts (number of heads, number of calls, etc.).
2. Choose the right discrete model (Bernoulli, Binomial, Poisson, …).
3. Write the PMF.
4. Check that probabilities add to 1.
5. Compute mean, variance, or required probability using the PMF.

What is a distribution?

Think of a distribution like a list that tells you how likely each event is.

What does discrete mean?

Discrete is just a fancy word for ‘separate’. The values do not glide smoothly; they jump from one value to another, like stepping‐stones.

Example (Like tossing coins)

If you toss a coin 3 times, you might get 0, 1, 2 or 3 heads. Those are four separate boxes. A discrete distribution writes down how much chance each box gets.

Add all chances and you get 1 (100 %).

So, a discrete distribution is just a table of chances where values are countable (0,1,2,…).

Below is a typical workflow to handle any discrete distribution question.

1. Define the random variable $X$
   Example: $X =$ “number of defective bulbs in a sample of 5”.

2. Select Model & Write PMF
   Since each bulb is good/defect, count of defects out of 5 → Binomial.

   $$P(X=k) = \binom{5}{k}p^{k}(1-p)^{5-k}$$

3. Probability Calculation
   Want $P(X \le 1)$? Add $P(X=0)+P(X=1)$.

4. Expectation & Variance
   For Binomial: $E[X]=np$, $Var(X)=np(1-p)$.

5. Check Logic
   Final probabilities must sit between 0 and 1; sums should be sensible.

This general five‐step ‘solution’ pattern works for any discrete distribution question.