statistical distributions

what is a random variable?

a variable whose value depends on the outcome of a random event

what is a sample space?

the range of values that a random variable can take

when is the variable discrete?

if it can take only certain numerical values

when is the variable random?

if it outcome is not known until the experiment is carried out

is a variable discrete or continuous?

it can be both

what is the probability distribution?

it fully describes the probability of any outcome in the sample space

how can the probability distribution for a discrete random variable be represented?

• probability mass function

• table

• diagram

here

what is discrete uniform distribution?

when the probabilities of different outcomes are the same

what is a probability mass function?

way of describing probability distribution

here

what does the probability for all outcomes of an event add up to?

1

how do you show the sum of the probability of all outcomes of an event?

Σ P(X = x) = 1

for a random variable x

what does X represent?

• random variable

• the number of successful trials

what does n represent?

number of trials (numtrial)

what does B represent?

that the random variable, X, is distributed binomially

what does p represent?

the probability of output

what does P represent?

the probability of the random variable X being chosen

how do you represent the number of successful trials?

random variable, X

what is the binomial distribution?

X ~ B (n, p)

• X = random variable

• B = distributed binomially

• n = numtrials

• p = probability of each output

when can you model X with a binomial distribution?

• fixed number of trials (n)

• 2 possible outcomes (success and failure)

• fixed probability of success (p)

• trials are independent of each other

P (success) = p

P (failure) = ? and why?

P (success) = p

P (failure) = 1 - p, because in binomial distribution there are only 2 possible outcomes and total probability is 1

how do you write the probability mass function if a binomial distribution, X ~ B (n, p) ?

here, and ncr

how else is ⁿC_r written?

here

what is the numtrial (n) sometimes called?

the index

what is the index?

the numtrial (n)

what is the probability (p) sometimes called?

the parameter

what is the parameter?

the probability (p)

what is the cumulative probability function?

tells you, for a random variable X, the sum of all the individual probabilities up to and including the given value x in P (X ≤ x) (i.e, the running count of probabilities)

what is the running count of probabilities for P (X ≤ x) known as?

the cumulative probability function

should your probability be equal use the symbol ≤ / ≥ or < / > ?

ALWAYS ≤ / ≥. if the function is in < / > , you must convert it

how do you convert P (X < 6)?

P (X < 6) → P (X ≤ 5)

explanation here with number line i cant be bothered rn

what does ‘greater than X’ mean?

X > x

what does ‘no more than X mean?

what does ‘at least X’ mean?

what does ‘fewer than X’ mean?

what does ‘at most X’ mean?