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What is probability & Mathematical Statistics?

It is the mathematical machinery necessary to answer questions about uncertain events.

Where scientists, engineers and so forth need to make results and findings to these uncertain events precise... Random experiment

“A random experiment is an experiment, trial, or observation that can be repeated numerous times under the same conditions... It must in no way be affected by any previous outcome and cannot be predicted with certainty.” i.e. it is uncertain (we don’t know ahead of time what the answer will be) and repeatable (ideally).The sample space is the set containing all possible outcomes from a random experiment. Often called S. (In set theory this is usually called U, but it’s the same thing)

Discrete probability Finite Probability This is where there are only ﬁnitely many possible outcomes. Moreover, many of these outcomes will mostly be where all the outcomes are equally likely, that is, uniform ﬁnite probability. An example of such a thing is where a fair cubical die is tossed. It will come up with one of the six outcomes 1, 2, 3, 4, 5, or 6, and each with the same probability. Another example is where a fair coin is ﬂipped. It will come up with one of the two outcomes H or T.

Terminology and notation. We’ll call the tossing of a die a trial or an experiment. Where we should probably reserve the word experiment for a series of trials, but when there’s only one trial under consideration, then the experiment is just the trial. The possible values that can occur when a trial is performed are called the outcomes of the trial. Following the notation in the text, we’ll generally denote the outcomes ω1, ω2,. .. ωn, where n is the number of possible outcomes. With a cubical die, the n = 6 outcomes are ω1 = 1, ω2

= 2, ω3 = 3, ω4 = 4, ω5 = 5, and ω6 = 6. With a coin, the n = 2 outcomes are ω1 = H, ω2 = T .Usually, we’ll introduce a symbol for the outcome of an experiment, such as X, and call this symbol a random variable.

So, if you ﬂip a coin and an H comes up, then X = ω1 = H, but if a T comes up, then X = ω2 = T. Let’s take the case of tossing a fair die. When it’s tossed, all six outcomes are equally probable, so they should all have the same probability. We’ll want the sum of all the probabilities of the outcomes to be 1, so for a fair die, each outcome will have probability 1/6.

We’ll use a couple of notations to express the assumption that each of the probabilities of the outcomes of a fair die is 1/6. Sometimes, to emphasize that we have a measure for each outcome, we’ll use the notation m(ω1) = m(ω2) = m(ω3) = m(ω4) = m(ω5) = m(ω6)=1/6.

But, more often, we usually use a random variable and a probability notation P (X=1) = P (X=2) = P (X=3)

= P (X=4) = P (X=5) = P (X=6) = 1/6. The probability notation is more convenient since we can use it to express probabilities other than X having a particular value. For example: P (X ≥ 5) expresses the probability that the die comes up with a value greater than or equal to 5, that is to say, either 5 or 6.

Since each of 5 and 6 have probabilities of 1/6, therefore P (X ≥ 5) = 2/6 =1/3. Thus, the probability notation allows us to express probabilities of sets of outcomes, not just single outcomes. The set of all possible outcomes is called a sample space. For tossing a fair die, the sample space is Ω = {ω1, ω2, ω3, ω4, ω5, ω6} = {1, 2, 3, 4, 5, 6}. But a sample space is more than just a set; it also has measures for each of its elements. That’s the function m : Ω →

[0, 1] which assigns to each outcome ωi its measure m(ωi). This function m is called a distribution function or a probability mass function. Sometimes we come across a diﬀerent symbol for it, like f. From m, we can derive probabilities for each subset E of the sample space Ω. A subset E of the sample space, that is, a set E of some of the outcomes,is called an event. The probability P (E) of an event E is deﬁned as the sum of measures of all the outcomes in the event. So, for example, we deﬁne the probability of the event E of the outcome being greater than or equal to 5 as

P (E) = P (X ≥ 5) = m(5) + m(6) =1/6 +1/6 =1/3.

Symmetry leads to uniform probabilities. So far, our two examples are uniform. A fair coin has two sides, and we assign the same probability to each because of the the physical symmetry of the coin. Likewise, we assign the same probability to each of the six sides of a fair die because of symmetry of the die.

There are lots of other situations where symmetry implies that the probabilities of the outcomes should be equal. For instance, some of the examples we’ll look at involve choosing balls from urns. Say an urn has 5 balls in it. Take out one of the balls at random. There are ﬁve possibilities, and each ball has the same probability of being chosen. Thus, we get a uniform probability where the probability of choosing any particular ball is 1/5. The symmetry of the situation is required, however, to conclude that the probability is uniform. For instance, if you throw a ball at the moon, there are two outcomes—either it hits the moon or it doesn’t. But the two outcomes are not symmetric, and it would be foolish to assume that the probability that your ball hits the moon is 1/2.

The frequency concept of probability.

When there’s no symmetry, we can’t assume the probability is uniform. Suppose you bend a coin. It’s no longer symmetric, so we can’t assume that H and T have the same probability. But you can get some idea of what the probability is by performing the experiment of ﬂipping the coin repeatedly. Suppose you ﬂip the coin 1000 times and you get 412 heads and 588 tails. It would be reasonable to conclude that P (H) is near 0.412, and P (T ) is near 0.588. However, it would be foolish to believe those are the exact probabilities. For that, we assume that the basic probabilities that we use are known. But for some of the situations to consider, we won’t know the basic probabilities and the problem will be to determined somehow by experiment to investigate what those probabilities are, and that’s the provision of Statistics.

Thus, the conclusion in the last paragraph that P (H) is near 0.412 is a statement in Statistics rather than a statement in Probability.…...

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