Probability and Statistics?

Question

I've been asking about thuis thing quite a few times, but I don't get usually get good answers.

Basically all I ask for is how to do the following and when do we use them?

standard deviation,

variance,

confidence interval,

density function,

binomial distribution,

normal distribution.

Also, if someone would like to teach me probability, it would be greatly appreciated.

Please explain thoroughly.

Answer 1

Standard deviation and variance are measures of spread or variation around the mean in a sample or population. The standard deviation is the square root of the variance.

A confidence interval is an interval estimate of the true population parameter. For example suppose we wish to estimate the unknown true population mean with 95% confidence and find the confidence interval to be 90 to 110. We conclude that we have 95% confidence that the true mean falls between 90 and 110 meaning that if we were to repeat our sampling procedure a very large number of times, then about 95% of all such constructed intervals would contain the unknown true mean.

Binomal and Normal distributions are two common probability distributions. The binomial distribution is a discrete distribution and is associated with experiments where each trial of the experiment is independent of each other trial, each outcome of the experiment is either a "success" or "failure" and the probability of a success is the same from trial to trial. The normal distribution is often used as an approximation to the true unknown underlying distribution. It is also very important from a theoretical point of view (e.g. the central limit theorem). It is a continuous bell-shaped symmetric distribution that has as its domain the real numbers and is characterized by two parameters, mu identifying the mean of the population and sigma identifying the standard deviation of the population.

The density (or mass for the discrete case) function is simply the mathematical function that relates for each x in the domain the probabilitiy associated with that x. For a continuous function that probability is of course 0. So we are more interested in probabilities like P(X < x). For discrete distributions we can calculate P(X = x) if needed. Note that a density (or mass) function f(x) has the properites that f(x) >= 0 for all x in the domain (or support) and when integrated (or summed for the discrete case) over all x in the support the result is 1.

To learn probability you first have to learn to count.

Math (and Stats) Rule!

Answer 2

There's really no way to explain all of these concepts in one answer. The topics you ask about would cover the better part of a math textbook.

I won't even attempt to explain all the stuff. My advice is that you pull out a textbook and go carefully over the proofs/derivations of the various formulas and concepts. Go over each proof line by line, seeking to understand *why* the author is doing what he's doing. Above all, resist the temptation to skip ahead if there's some line that you don't fully understand.

If you want specific answers on individual topics, split them up into separate questions and ask them one at a time. Seriously, though, you don't need our answers: the stuff you need is explained perfectly well in your textbook. You just need to focus on understanding the proofs, rather than simply skipping ahead to the formulas and then trying to apply them to problems.

Students are always tempted to skip over the proofs, but they're really the most important part of any math textbook. For me at least, the best way to learn a concept is by understanding the proof.

I'm sorry if this advice doesn't seem terribly helpful. I know it doesn't provide you with a quick and easy fix. However, understanding ultimately has to come from your own effort, there's no shortcut for that.