What should I review for an introduction Probability Course

[MidgetDwarf]

What should I review for an introductory probability course? Here is the course description.
Math 224 I believe is Calculus 3 at this school( multivariable).

380. Probability and Statistics (3)
Prerequisite: MATH 224.
Frequency interpretation of probability. Axioms of probability theory. Discrete probability and combinatorics. Random variables. Distribution and density functions. Moment generating functions and moments. Sampling theory and limit theorems.
Letter grade only (A-F). (Lecture 3 hrs.) Not open for credit to student with credit in STAT 380.

 

[axmls]

It doesn't hurt to review some set theory and some common sums, as well as integration techniques.

 

[chiro]

Hey MidgetDwarf.

I'd recommend getting used to basing your intuition of probability on the Kolmogorov axioms instead of trying to use intuition.

Probability is difficult because many try and use a combination of intuition and an attempt to fit formulas in ways that don't fit.

If you can get probability from a first principles approach that clarifies the symbolic nature of probability with something intuitive then it should be a lot easier.

It will also help understand the more complicated results and models in probability and you will be able to know why the formulas work and more importantly when to use them.

 

[MidgetDwarf]

Hey MidgetDwarf.

I'd recommend getting used to basing your intuition of probability on the Kolmogorov axioms instead of trying to use intuition.

Probability is difficult because many try and use a combination of intuition and an attempt to fit formulas in ways that don't fit.

If you can get probability from a first principles approach that clarifies the symbolic nature of probability with something intuitive then it should be a lot easier.

It will also help understand the more complicated results and models in probability and you will be able to know why the formulas work and more importantly when to use them.

Can you elaborate on your post please? I am new to probability. This is my first exposure to the subject.

 

[chiro]

I can elaborate on this.

The Kolmogorov axioms are the basic axioms for defining what valid probabilities are.

Basically they look at probabilities being in between 0 and 1, how to add probabilities [P(A OR B) = P(A) + P(B) - P(A and B)], and all probabilities add to 1.

With probability you often get a lot of formulas for specific kinds of processes and a lot of results.

What tends to happen is for people new to probability is that they try and "fudge" the results by looking for formulas and using them in ways that aren't understood.

If you understand the probability axioms properly, then it means you can not only understand the derivation of the formulas but more importantly when they should be used.

Probabilities form distributions and you can combine different distributions together as well as finding subsets (or conditional distributions) and the intuition for all of the conditional probabilities should be based on finding a "slice" of a distribution and treating that as if it were its own distribution.

Once you understand how to think of probability in terms of the basic Kolmogorov axioms and look at probabilities in terms of events and not trying to find the formula to use in answering a question, then it means that you can understand what you are doing and as above - know how and why to use a formula if you can use it.

You will find that as your education progresses you will need to derive results that others haven't already figured out and you can't do that without a proper understanding of the basic axioms, developing intuition through things like tree diagrams and other techniques, and understanding all the different probability formulas and what they mean along with their derivation.

If you can work with the raw events then you will be doing a lot better than most students in my opinion because they try and use their intuition when it screws it up.

Aside from this there are the mathematical tools of linear algebra and multivariable calculus that you will need to know. They are tools and can be likened to a carpenters tools (like drills and bits) to make furniture or other forms of produced goods.

The ideas in calculus need to be understood to make sense of what they mean in a probabilistic context. This will involve knowing what optimization, differentiation and integration are doing and how that relates to the particular result. Optimization is a common technique when trying to estimate probabilities and do statistics.

Aside from this, the only other significant advice is to work on the problems with the above in mind.

Many problems will force you to find out what the events are, what the distribution is, and how to convert all the information you are given to mathematical statements which are combined together to get some "output" set of constraints which represents a solution of some sort.

Doing this takes a bit of practice and it's a process of being able to take bits of information and translate them into mathematics. The more you do problems the easier this becomes and a good piece of advice regarding this is to think about exhausting all possibilities (in terms of the information you need to make a resolution to your question) and find out if the question can provide it.

If you find it can't then you will need to assume things and learning how to do this will come through experience.

If you can put all of this into context then I think you will have an edge when it comes to your coursework because at least you will understand how all of it fits in with not only understanding the subject, but being able to take a problem and work towards its resolution.