Stats exam 2

Random Experiment

A trial whose possible outcomes are known, but actual outcomes are unknown. For example. flipping a coin.

Probability

A long run proportion of an outcome in a repeated trial. For example, the odds of flipping a coin and getting heads are 50%.

Law of large numbers

In a repeated trial, a long run proportion of an outcome approaches its probability.

Sample space (S):

A set of all possible outcomes from an experiment. For example, the sample space for flipping a coin would be S = {Heads, Tails}.

Event (A,B,C,…..)

Any subset of S. For example, A={Heads}.

Equally likely outcome experiment

An experiment in which the results are equally likely to occur.

Probability Rule

For any A, 0<P(A)<1. 0 means it won’t, and 1 means it will.

When is something unusual?

If P(A)<0.05

Compound event

an event that is formed by more than one event.

1 OR 2

1,2 or both will occur

1 AND 2

Both will occur together.

General Addition Rule

P (A or B) is P(A)+P(B)-P (A and B)

Mutually Exclusive

If event A and B cannot occur at the same time (P(A and B)=0)

Compliment of A.

Event that makes A not occur

Conditional Probability

a probability that only applies with certain context

Independence

2 events are celled independent if the occurrence of one event does not affect the probability of the other events

Random Variable (R.V.) = X₁,X₂…..

Numerical outcome of a random experiment.

Mean of a discrete R.V. X.

μ_x= Sum (X*P(X))

The mean is also called…..

Expected value.

μ_{x is……}

The long-run average of outcomes from repeated trials.

Variance of a discrete R.V. X.

Sum of (x-μ_x)^₂P(x) or Sum of X²p(x)-μ_x²

Standard deviation of R.V

sqrt(varience²)

Binomial Experiment

Fixed number of trials, each trial has two outcomes, the probability is the same, and trials are independent of each other.

Binomial Experiment example.

Suppose 10 people enter a prize contest, where the prob. of winning is 0.1. Let x be the number of winners among those 10 people.

Binomial probability distribution example:

n=3 (trials) and x=2 (successes).

In general, P(X=X)=

_nC_xP^x(1-P)^n-x

_nC_x

n!/x!(n-x)!

_nC_x example

₃C₂=3!/2!(3-2)!=3

Normal Distribution

most widely used continues distribution.

Probability Density curve

Provides the proportion of a population that falls between a and b as the area under the curve between a and b.

Uniform Distribution

All values in an interval are equally likely.

Area under curve=

1

if X intercept = A

then Y intercept = 1/A

If area under the curve is 1, then……..

area=probabilty.

bell shaped

clustered around average

symmetric

mirror images around the center.

(Central Limit Theorem) For large n (n>30),

The sampling distribution of xbar is approximately a normal distribution.

Proportion of 1s

if x= 0 or 1, 1’s/total.

Continuity correction

either add 0.5 or subtract 0.5