In this section, we will only be dealing with discrete probability distributions.

Discrete probability distributions deal with discrete random variables. We will use simple examples where the sample space is clear, so that it becomes easy to calculate the probability… that is, if you know what you are doing 😉.

 

What the Heck is a Probability Distribution?

Say you have a discrete random variable X with sample space { x_{1}, x_{2}, x_{3}, \ \text{...} \ , \ x_{n}}} }, with corresponding probabilities { p_{1}, p_{2}, p_{3}, \ \text{...} \ , \ p_{n}}} }, where P(X = x_{i}) = p_{i}  for i = 1, 2, 3, …, n. Then { p_{1}, p_{2}, p_{3}, \ \text{...} \ , \ p_{n}}} } describes to probability distribution of X. (Here we are removing the word “discrete” because ALL of the probabilities that we are dealing with in this section will be discrete).

Wow. That was a lot of symbols! What does that all mean?

Examples are good! Let’s go back to our trusted coin example.

Say you have a fair coin. We define the discrete random variable X as the number of heads facing upwards when tossing the coin twice.

What is our first step? We need to figure out the sample space for X. Since X is the number of heads facing upward when tossing the coin twice, the sample space is {0,1,2} because the only possibilities is for both throws to have no heads (0), one head and one tail (1) or both heads (2).

Next, we look for the corresponding probabilities. This is easier done if we list ALL possible outcomes for tossing a coin twice:

TT, HT, TH, HH

There are 4 possible outcomes. There is only one option for no heads (TT), so the corresponding probability for x_1 = 0 is 1/4. There are two outcomes for one head (HT and TH), so the corresponding probability for x_2 = 1 is 2/4 or 1/2. There is only one option for two heads (HH), so the corresponding probability for x_2 = 2 is 1/4.

Putting this together, the corresponding probabilities are {1/4, 1/2, 1/4}. This describes the probability distribution.

 

Important notes on Probability Distributions

  1. All of the probabilities need to be in between 0 and 1. ie: 0 \leq p_i \leq 1 for all i = 1 to n
  2. When you add ALL of the probabilities in the probability distribution, it needs to add to 1. ie: $$\sum_{i=1}^{n}p_i = p_1+p_2+. \ . \ .+p_n=1$$

Number one is fairly obvious. a probability of zero means the particular event will never happen. Can you think of what a negative probability would mean? Yeah, me neither. So 0 is the natural lower limit cutoff. A probability of 1 means that a particular event is 100% certain of happening. There is nothing better than 100% certain of happening, no matter what advertisements try telling you.

All number two is telling you is that if you add up ALL of the probabilities corresponding to ALL of the events, then you have covered 100% of the possibilities. Having one of the events occur is 100% certain of happening. This should be obvious because you took into account all of the events.

 

Notation

It may be useful to know how to communicate the mathematical notation in words.

NotationHow to say it in words
P(X=3)The probability that X equals 3
P(X<3)The probability that X is less than 3 (or strictly less than 3)
P(X≤3)The probability that X is at most 3 (or less than or equal to 3)
P(X>3)The probability that X is more than 3 (or X is strictly greater than 3)
P(X≥3)The probability that X is at least 3 (or X is greater or equal to 3)
P(3The probability that X between 3 and 7
P(3≤X≤7)The probability that X is al least 3 but no more than 7
P(3The probability that X is more than 3 but no more than 7
P(3≤X<7)The probability that X is at least 3 but less than 7