Discrete Probability and Distribution to understand the topic.

[OhyesOhno]

I'm currently studying this topic at school... I'm a 12th grader. So I just want to ask, does anyone know how to comprehend the topic in an easier way? or probably anyone know a guide somewhere in the net?

I am just confused at binomial distribution, and poisson. I understand a little bit about variance and mean... but that's about it...

 

[statdad]

Your question is rather vague. Here are some general ideas.

• In a binomial probability setting we think of repeating some task a specified number of times. After each performance we examine the result to determine whether we've had a success or a failure (generic terms). For example, roll a pair of fair dice 20 times, each time looking to see whether the sum of the top dots is 8 (that's our success) or is something else (anything else is our failure)
• In a Poisson problem there is no fixed number of times an experiment is performed, but we are counting the number of times something (we call it a rare event ) occurs.
• In both binomial and Poisson problems, a distribution function is used to calculate the probability of some outcome. If we want the probability that the sum of 8 occurs at least 12 times over the 20 rolls, the distribution function can be used to find that.
• For binomial, and for Poisson, there are some technical assumptions that must be made about the situation you are studying in order to be sure that the mathematics you use fits the situation. You can usually find very good, brief, descriptions of those assumptions, and their practical implications, in texts.

As I said, your request was rather vague - is this along the lines you were seeking?

 

[OhyesOhno]

Yeah that answered some of the questions I have. I just want to know some terms and definitions... like:

- Random variables
- What is P(X = x)?
- Probability distribution of a random variable?

Thanks again! Oh, and I might be asking more questions later, I hope you wouldn't mind :)

 

[statdad]

I've tried to be intuitive with these comments - rather like the first comments I give my students at the beginning of probability.
- Random variables: The ''variable'' part of the name indicates that you are studying something that isn't constant. The ''random'' part tells you that you are studying a variable whose values are governed by some type of probability rule. Here is an extremely simple example. Suppose we have a fair die - a die for which all six sides are as equal in area and weight as can be. Each side is marked with either 1, 2, 3, 4, 5 or 6 dots. An obvious random variable for this setting is the number of dots the side that is on top after a roll of the die. The number of dots will vary from roll to roll, and since it is impossible to precisely predict which side will come up, the side is random.
- What is P(X = x)? - You will typically see capital (upper-case) letters used as names for random variables. In my die example above I could say "Let X be the number of dots on the top side of the die after a single roll". Similarly, you will see lower-case letters in formulas used to indicate specific values of a random variable. Again, in my example, if I wanted to know the probability that after one roll the side with 4 dots would be up, I am asking for $$\Pr(X = 4)$$. This leads to your next question.
- Probability distribution of a random variable? Every random variable has a probability distribution. Probability distributions tell us the rules for calculating probability for random variables. A distribution may be a table, or a formula, or be described by areas underneath curves (calculus is needed for the final situation).
General comments about probability distributions.

• Every probability distribution contains information that allows you to calculate any probability that relates to the associated random variable
• The total probability associated with a distribution is 1 (100%)
• Every probability you, or anyone, calculates, is between 0 and 1.

Think of my die example one final time. It seems reasonable to assume the sides all of the same chance of being on top when the die is rolled. The probability distribution can be written as a table this way:

$$\begin{tabular}{lcccccc} X & 1 & 2 & 3 & 4 & 5 & 6 \\ p(x) & 1/6 & 1/6 & 1/6 & 1/6 & 1/6 & 1/6 \\ \end{tabular}$$

What is the probability that the side with three dots comes up? it is

$$\Pr(X = 3) = p(3) = \frac 1 6$$

What is the probability the number of dots will be two or fewer?

$$\Pr(X \le 2) = p(1) + p(2) = \frac 1 6 + \frac 1 6 = \frac 2 6 = \frac 1 3$$

Finally, it is possible to write this same distribution as a function. It is

$$p(x) = \frac x 6, \quad x = 1, 2, 3, 4, 5, 6$$

The question of which is preferred depends on the people using it.

Hope this helps.

 

[statdad]

MAJOR BRAIN FART on my part. I finished my previous post this way:

Finally, it is possible to write this same distribution as a function. It is

$$p(x) = \frac x 6, \quad x = 1, 2, 3, 4, 5, 6$$

The question of which is preferred depends on the people using it.

Hope this helps.

Answers just don't get much more wrong than this - sorry. The correct way to write the result is this:

$$p(x) = \frac 1 6, \quad x = 1, 2, 3, 4, 5, 6$$

- i.e. - the probability of obtaining $$x$$ dots on a roll is always $$1/6$$.

 

[OhyesOhno]

okay thanks a lot!