Starnes Statistics Unit 4-Probability, Random Variables, and Probability Distributions

Random process

A process that generates outcomes that are determined purely by chance.

Probability

The chance that a random process will occur, which is a number between 0 and 1 that describes the proportion of times the outcome would occur in a very long series of trials.

law of large numbers

The more we observe trails of any random process, the closer it will reach its calculated probability.

simulation

A model that imitates a random process in such a way that simulated outcomes are consistent with real world outcomes

Probability model

A description of some random process that consists of two

sample space

The list of all possible outcomes

finding probabilities with equally likely outcomes

If all outcomes in the sample space are equally likely, the probability that event A occurs can be found using the formula, P(A)=(number of outcomes in event A)/(total number of outcomes in the sample space)

event

Any collection of outcomes from some random process.

A^C/Complement

The probability when A doesn't exist

Complement rule

P(A^C)=1-P(A).

General addition rule

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

mutually exclusive events

When two events have no outcomes in common and P(A and B)=0

Addition rule for mutually exclusive events

When events A and B are mutually exclusive, P(A or B)=P(A)+P(B)

Venn diagram

A diagram instating of one ore more circles surrounding by a rectangle. Each circle represents an event, and the area inside the circle represents the sample space of the probability.

Intersection

All the points where the two events meet, which is called P(A and B) and denoted as A ∩ B

Union

All the points in both of the events, which is called P(A or B) and is denoted as A ∪ B

Conditional probability

The probability that one event happens given that another event is known to have happened is called a conditional probability. The event that A happens given B is denoted as P(A/B)

Formula for conditional probabilities

P(A/B)=P(A and B)/P(B)=P(A ∩ B)/P(B)=P(both events occur)/P(given event occurs)

Independent events

Two events that if knowing whether or not one has occurred doesn't change the probability that the other event will happen. Thus events are independent if P(A/B)=P(A/B^C)=P(A), or P(B/A)=P(B/A^C)=P(B)

General multiplication rule

For any random process, the probability that events A and B occur can be found using the general multiplication rule P(A and B)=P(A ∩ B)=P(A)*P(B/A)

Multiplication rule for independent events

For any two independent events, the probability that A and B both occur can found by the following formula: P(A and B)=P(A ∩ B)=P(A)*P(B)

Tree diagram

A diagram that shows the sample space and probability of a random process involving multiple stages. Note that the probabilities are always conditional probabilities.

Random variable

The variable that takes numerical values that describe the outcomes of a random process

probability distribution

The graph that gives a random variable's possible values and probabilities.

discrete random variable

A random variable that takes a fixed set of possible values with gaps in between them. Discrete random variables result from counting something

n (probability distribution)

number of trials

x(i) (probability distribution)

value being calculated for its probability

p(i) (probability distribution)

probability of the value's success

Mean (expected value) of a discrete random variable

The average value over many trails of the same random process

Mean (expected value) of a discrete random variable formula

Sum(x(i)p(i))

Standard deviation of a discrete random variable

Number that measures how much the values of the variable typically vary from the mean after many trials of the random process

Variance of a discrete random variable

Sum(p(i)*(x(i)-mean)^2)

standard deviation of a discrete random variable

sqrt(Sum(p(i)*(x(i)-mean)^2))

continuous random variable

Type of random variable that can take any value in an interval on the number line. These variables usually come from measuring something.

The probability distribution of a continuous random variable

a density curve

How to find probabilities for a continuous random variable

The probability of any event involving a continuous random variable is the area under the density curve and directly above the values on the horizontal axis that make up the event

Independent random variables

Variables that knowing the value of one variable doesn't help us predict the other.

standard deviation of the sum of two independent random variables formula

sqrt(stddev(X)^2+stddev(Y)^2)

standard deviation of the difference of two independent random variables formula

sqrt(stddev(X)^2+stddev(Y)^2)

binomial setting

When we perform n independent trials of the same random process and count the number of times that a particular outcome occurs

Binomial random variable

The count of success (X) in a binomial setting, where the possible values are positive (or 0) integers

Binomial distribution

The distribution of x that identifies the number of trials (n) and the probability of the success (p)

binomial coefficient

(n (next line) x) = n!/(x!*(n-x)!), for x being an integer greater than or equal to 0.

Binomial probability formula

The probability of getting exactly x successes in n trials is P(X = x) = binomial coefficient*p(x)*(1-p)(x)

Mean (expected value) of a binomial random variable formula

n(i)*p(i)

Standard deviation of a binomial random variable

sqrt(n(i)*p(i)(1-p(i)))

10% condition

When taking a random sample (n) from a population (N), we can treat individual observations as independent when performing calculations, as long as n

Large counts condition

The probability distribution of X is approximately normal if np ≥ 10 and n(1-p)≥10.

Geometric setting

When we perform independent trials of the same random process and record the number of trials it takes to get one success. The probability must be the same on each trial

Geometric random variable

The number of trials it takes to get a success in a geometric setting, which are positive integers

Geometric distribution

The probability distribution of a geometric random variable, with a probability of success on each trial

Geometric probability formula

P(X = x)=p(i)(1-p(i))^(x-1)

Mean of a geometric random variable

1/p(i)

standard deviation of a geometric random variable

(sqrt(1-p(i)))/p(i)