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UA - ST 260 - Study Guide - Midterm

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ST 260 EXAM 2

Study Guide

Note: U = intersection, is upside down U = union

Conditional Probability

is the probability of event A will occur, given that event B has already occurred. P(A | B) = P(A U B) / P(B)

We assume that P(B) does not equal 0

-For example,

A= “it rains tomorrow” B= “it rains today”

P(B)=0.8 P(A | B)= 0.9

What is the chance it will rain both days?

P(A U B) =P(B) x P(A | B) = (0.8) x (0.9) = 0.72

Disjoint Events (mutually exclusive events)

this is when A and B are two exclusively different events

if two events A and B are mutually exclusively, the probability of the union of A and B equals the sum of the probabilities of A and B; that is, P(A U B) = P(A) + P(B)

For example,

You toss a coin 3 times

S = {HHH,HHT,HTH,THH,HTT,THT,TTH,TTT}

P(HHH) = 1/8

P(THT) = 1/8

P(HHH U THT)= 1/8 + 1/8 = 2/8

Independent Events

Events A and B are independent events if the occurrence of B does not alter the probability that A has occurred; that is, events A and B are independent if

P(A | B) = P(A)

When events A and B are independent, it is also true that

P(B | A) = P(B) or P(A U B) = P(B) x P(A)

Events that are not independent are said to be dependent.

For example, toss a coin 2 times

Event A = {HEAD on the 1st toss}

Event B = {HEAD on the 2nd toss}

P(A | B) = P(A U B) ÷ P(B) =1/4 ÷ ½ P(A

P(A | B) =P(A) thus, events are independent.

Discrete Probability Distributions

-Random variable: a variable that can take on different numeric values according to the outcome of an experiment.

Example: Toss a fair die

X= # spots S= (1, 2, 3, 4, 5, 6)

Example: Toss a coin

S= {H, T} X= # of heads

X is the random variable in both examples

-Discrete random variable: a random variable whose possible values are associated with a countable set of numbers. If you want to learn more check out What are the types of archaeology?

Examples:

1. The number of sales made by a salesperson in a given week: x =0,1,2, …

2. The number of consumers in a sample of 500 who favor a particular product over all competitors: x= 0,1,2,….,500

3. The number of errors on a page of an accountant’s ledger: x= 0,1,2,….

4. The number of customers waiting to be served in a restaurant at a particular time: x=0,1,2,…..

Intuitive Interpretation

If we repeat an experiment many, many times under the same conditions, and if we average the results, then the average will represent the expected value.

Bernouilli Distribution

“X” is a discrete, binary random

Binary means there can only be two possible outcomes, either 0 or 1 Example: Take one card at random. Success is picking a “diamond.” If “diamond” appears, X=1 If “no diamond” appears, X=0 We also discuss several other topics like one of the most common editing techniques designed to hide the instantaneous and potentially jarring shift from one camera viewpoint to another is:

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Bernouilli only has one parameter: p= the probability of success We also discuss several other topics like sta 13 study guides

Binomial Distribution

X is a discrete random variable If you want to learn more check out psy 350 exam 3

X= a count of the number of successes in n independent Bernouilli trials It has to have:

1. A fixed number of trials, n

2. All n trials must be independent of each other

3. The probability of success remains the same for all trials Binomial has two paramaters:

n = the fixed number of trials

p=the probability of success for each trial

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