quick memorize formulas

Question 1

When do we use hypergeometric and what formulas do we use?

Answer 1

The hypergeometric distribution is one unlike any of the others. There are 3 conditions which must be fulfilled:
Trials that are not identical
Trials that are dependent
Two possible outcomes; success or failure

Hypergeometric distribution probability formula

P(x) = (aCx)(n-aCr-x)/nCr

where n = # of possible outcomes
a = # of successful outcomes r = # of dependent trials
x = number of required successes

Hypergeometric distribution expected value = probability of success times the number of dependent trials.

E(x) = r (a/n)

Question 2

P(x) = (aCx)(n-aCr-x)/nCr  
E(x) = r (a/n)
Trials that are not identical
Trials that are dependent 
Two possible outcomes; success or failure

Answer 2

hypergeometric distribution

Question 3

Normal approximation

Answer 3

Standard deviation can be calculated by 𝜎=√(npq)

Condition for normal approximation = If X is a binomial random variable of n independent trials, each with probability of success p, and if np>5 and nq>5. This lets the binomial random variable be approximated by a normal distribution.

Question 4

𝜎=√(npq)

np>5 and nq>5

Answer 4

normal approximation

Question 5

geometric distributions

Answer 5

Wait time

Example: Determine the probability of producing 3 cars before finding a defect if the probability of a defect is 1%.

The formula for geometric distribution probability is:
P(x) = (q^x) (p) where x is the number of failures before the first success, p is the probability of success and q is the probability of failure

P(x) = (q^x) (p)
= 0.99^3 (0.01)
=0.0097
The probability is 0.0097 or 0.97% chance of finding a defect

Expected value
E(x) = q/p

Question 6

P(x) = (q^x) (p)
E(x) = q/p
Wait time

Answer 6

geometric distributions

Question 7

Binomial distributions

Answer 7

P(x) = nCx (p^x) (q^n-x)
n = # of trials			p = probability of successes 
x = # of successes 		q = probability of failure 

E(x) = np where n = number of trials and p = probability of success

Bernoulli Trials, they consist of:
Are the trials identical?
Trials have to be done the same way each time.
Are the trials independent?
The first outcome of a trial does not impact the next.
Do the trials have two outcomes (success or failure)?
The desired outcome will either happen (success) or not happen (failure)

Question 8

P(x) = nCx (px) (q n-x)
n = # of trials			p = probability of successes 
x = # of successes 		q = probability of failure 

E(x) = np where n = number of trials and p = probability of success

Bernoulli Trials, they consist of:
Are the trials identical?
Trials have to be done the same way each time.
Are the trials independent?
The first outcome of a trial does not impact the next.
Do the trials have two outcomes (success or failure)?
The desired outcome will either happen (success) or not happen (failure)

Answer 8

Binomial distribution

Question 9

Uniform distribution

Answer 9

When all outcomes in a distribution are equally likely in any single trial, we call this a uniform probability distribution.
P(X) = 1 / n

Expected value - an expectation/expected value, E(X), is the predicted average of all possible outcomes in a single trial of a probability experiment. Expected value is the long-run average.

Formula E(X) = sum of x P(x)

Question 10

P(X) = 1/n
E(X)= Sum of xP(x)
All outcomes in a distribution are likely

Answer 10

uniform distribution