STAT chapter 4

law of large numbers

Draw independent observations at random from any population with fi- nite mean μ. Decide how accurately you would like to estimate μ. As the number of observations drawn increases, the mean x of the observed val- ues eventually approaches the mean μ of the population as closely as you specified and then stays that close

complement

The complement Ac of an event A consists of exactly the outcomes that are not in A. Events A and B are disjoint if they have no outcomes in common. Events A and B are independent if knowing that one event occurs does not change the probability we would assign to the other event.

normal distribution

The density curves that are most familiar to us are the Normal curves. Because any density curve describes an assignment of probabilities, Normal distribu- tions are probability distributions. Recall that N(μ, σ ) is our shorthand for the Normal distribution having mean μ and standard deviation σ . In the language of random variables, if X has the N(μ,σ) distribution, then the standardized variable Z=X−μ σ is a standard Normal random variable having the distribution N(0, 1).

intersection

The intersection of any collection of events is the event that all of the events occur. P(A and B and C) = P(A)P(B | A)P(C | A and B)

varience

The variance σ2X is the average squared deviation of the values of the variable from their mean. For a discrete random variable, σX2 =(x1 −μ)2p1 +(x2 −μ)2p2 +···+(xk −μ)2pk

general addition rules for uniont

For any two events A and B, P(A or B) = P(A) + P(B) − P(A and B)

disjoint

If A and B are disjoint, the event {A and B} that both occur has no outcomes in it. This empty event is the complement of the sample space S and must have probability 0. So the general addition rule includes Rule 3, the addition rule for disjoint events.

equally likely outcomes

If a random phenomenon has k possible outcomes, all equally likely, then each individual outcome has probability 1/k. The probability of any event A is P(A) = count of outcomes in A count of outcomes in S = count of outcomes in A k

rules for means

Rule 1. If X is a random variable and a and b are fixed numbers, then μa+bX =a+bμX Rule 2. If X and Y are random variables, then μX+Y =μX +μY

independent trials.

That is, the outcome of one trial must not influence the outcome of any other.

mean

The mean μ is the balance point of the probability histogram or density curve. If X is discrete with possible values xi having probabilities pi, the mean is the average of the values of X, each weighted by its probability: μX =x1p1 +x2p2 +···+xkpk

random

We call a phenomenon random if individual outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a large number of repetitions.

conditional probabilty

When P(A) > 0, the conditional probability of B given A is P(B|A)= P(AandB) P(A)

variance

(xi −μX)2* Pi A basic numerical de- scription requires in addition a measure of the spread or variability of the dis- tribution. The variance and the standard deviation are the measures of spread that accompany the choice of the mean to measure center. Just as for the mean, we need a distinct symbol to distinguish the variance of a random variable from the variance s2 of a data set. We write the variance of a random variable X as σX2. The standard deviation σX of X is the square root of the variance.

probability rules

1. Anyprobabilityisanumberbetween0and1.Anyproportionisanumber between 0 and 1, so any probability is also a number between 0 and 1. An event with probability 0 never occurs, and an event with probability 1 occurs on every trial. An event with probability 0.5 occurs in half the trials in the long run. 2. All possible outcomes together must have probability 1. Because every trial will produce an outcome, the sum of the probabilities for all possible outcomes must be exactly 1. 3. If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities. If one event occurs in 40% of all trials, a different event occurs in 25% of all trials, and the two can never occur together, then one or the other occurs on 65% of all trials because 40% + 25% = 65%. 4. The probability that an event does not occur is 1 minus the probability that the event does occur. If an event occurs in (say) 70% of all trials, it fails to occur in the other 30%. The probability that an event occurs and the probability that it does not occur always add to 100%, or 1 Rule 1. The probability P(A) of any event A satisfies 0 ≤ P(A) ≤ 1. Rule 2. If S is the sample space in a probability model, then P(S) = 1. Rule 3. Two events A and B are disjoint if they have no outcomes in common and so can never occur together. If A and B are disjoint, P(A or B) = P(A) + P(B) This is the addition rule for disjoint events. Rule 4. The complement of any event A is the event that A does not oc- cur, written as Ac. The complement rule states that P(Ac) = 1 − P(A)

continous random variable

A continuous random variable X takes all values in an interval of num- bers. The probability distribution of X is described by a density curve. The probability of any event is the area under the density curve and above the values of X that make up the event.

probabilty model

A de- scription of a random phenomenon in the language of mathematics is called a probability model. This description of coin tossing has two parts: • A list of possible outcomes • A probability for each outcome

discrete random variable

A discrete random variable X has a finite number of possible values. The probability distribution of X lists the values and their probabilities: Value of X x1 x2 x3 · · · xk Probability p1 p2 p3 · · · pk The probabilities pi must satisfy two requirements: 1. Every probability pi is a number between 0 and 1. 2. p1 + p2 + · · · + pk = 1. Find the probability of any event by adding the probabilities pi of the particular values xi that make up the event.

random variable

A random variable is a variable whose value is a numerical outcome of a random phenomenon.

disjoint and independent rules

The multiplication rule P(A and B) = P(A)P(B) holds if A and B are indepen- dent but not otherwise. The addition rule P(A or B) = P(A) + P(B) holds if A and B are disjoint but not otherwise. Resist the temptation to use these simple formulas when the circumstances that justify them are not present. You must also be certain not to confuse disjointness and independence. Disjoint events can- not be independent. If A and B are disjoint, then the fact that A occurs tells us that B cannot occur

conditional probability

The new notation P(A | B) is a conditional probability. That is, it gives the probability of one event (the next card dealt is an ace) under the condition that we know another event (exactly 1 of the 4 visible cards is an ace). You can read the bar | as "given the information that."

probability

The probability of any outcome of a random phenomenon is the propor- tion of times the outcome would occur in a very long series of repetitions. Probability describes what happens in very many trials, and we must actu- ally observe many trials to pin down a probability. The idea of probability is empirical. Simulations start with given probabili- ties and imitate random behavior, but we can estimate a real-world probabil- ity only by actually observing many trials

s

The sample space S of a random phenomenon is the set of all possible outcomes.

the multiplication rule for independent events.

Two events A and B are independent if knowing that one oc- curs does not change the probability that the other occurs. If A and B are independent, P(A and B) = P(A)P(B) This is

An event

is an outcome or a set of outcomes of a random phenomenon. That is, an event is a subset of the sample space.