Ch. 6 Stats Vocab

probability model/distribution

describes the possible outcomes of a chance process AND the likelihood of those outcomes will occur

• can use a table or tree diagram

• sum of all probabilities must equal 1

• every probability is between 0 and 1, inclusive

random variable

takes numerical values that describe the outcomes of a chance process

• values of the sample space

expected value/mean of a discrete random variable

an average of the possible outcomes, but a weighted average in which each outcome is weighted by its frequency

• does not have to be a possible outcome; decimals are fine

calculating expected value of discrete variable

μ_x = E(X) = (x₁)(p₁) + (x₂)(p₂) + (x₃)(p₃) + …

• make sure to show at least three terms

  • first two terms and the last term

median of a discrete random variable

smallest value for which the cumulative probability equals or exceeds 0.5

variance of expected value or a discrete random variable

Var(X) = σ_x² = (x₁ - μ_x)²p₁ + (x₂ - μ_x)²p₂ + (x₃ - μ_x)²p₃ + …

• make sure to show at least three terms

  • first two terms and the last term

standard deviation of expected value or discrete random variable

the measure of variability for the center of a discrete random variable

σ_x = [(x₁ - μ_x)²p₁ + (x₂ - μ_x)²p₂ + (x₃ - μ_x)²p₃ + …]^1/2

• square root of the variance

adding or subtracting a constant

• does not change the shape

• does not change the measure of variability

• adds/subtracts the constant to the measure of center and location of each point

  • mean is different, but standard deviation is the same

multiplying or dividing a constant

• does not change the shape

• multiples/divides measure of center and location of each point by the constant

• multiples/divides the measure of spread by the constant

  • both mean and standard deviation change

    • mean + standard deviation = multiplied by the constant

    • variance = multiplied by the constant squared

interpretation of expected value and standard deviation

On average, the (variable) varies from the mean of "x,” by about “y” (units).

independent random variables

when knowing the value of X does not help predict the value of Y

rule for sum of independent variables

mean —> μ_sum = μ_x+y = μ_x + μ_y

standard deviation —> σ_sum = σ_x+y = (σ_x² + σ_y²)^1/2

• aka square root of the sum of the variances

rule for difference of independent variables

mean —> μ_diff = μ_x-y = μ_x - μ_y

standard deviation —> σ_diff = σ_x-y = (σ_x² + σ_y²)^1/2

• aka square root of the sum of the variances

• still ADDING

• only for independent variables

continuous random variable

can take any value in an interval on the number line

probability for continuous random variable

the area under the density curve and directly above the values on the horizontal axis that make up the event

probability of a single outcome of continous variable; P(x = a)

is always 0

• probability of continous variables are always a range of values

combining two independent normal variables

any sum or difference of independent normal variables is also normally distributed

additive transformations

“y” amount of trial

mean —> μ_{x¹+x²+…+x^y} = y × μ_x

standard deviation —> σ_{x¹+x²+…+x^y} = [y × σ_x²]^1/2

multiplicative transformations

“y” amount of trial

mean —> μ_yx = y × μ_x

standard deviation —> σ_yx = [y² × σ_x²]^1/2 = y × σ_yx

combination

counting when order is not considered

permutation

counting when different orders is considered

multiplication rule

if one event can occur in “m” ways, a second event in “n” ways, and a third event in “r” ways, then the three events can occur in “m × n "× r”

• think of a tree diagram

repetition of an Event

if one event with “n” outcomes occurs “r” times with repetition allowed, then the number of ordered arrangements is “n^r”

calculating permutations

nPr = n!/(n - r)!

• n = number of objects

• r = number of positions

calculating combinations

nCr = nPR/r! = n!/[r! × (n - r)!]

• n = number of objects

• r = number of positions

• number of permutations/arrangement of “r” objects

binomial setting

when we perform “n” independent trials of the same chance process and count the number of times that a particular outcome occurs

how to tell if binomal

BINS

• B = Binary, can be classified as “success” or “failure”

• I = Independent, knowing the outcome of one trial must not tell anything about the outcome of the next

• N = Number, the number of trials “n” has already been fixed in advance

• S = Same probability, same chance of success of “p” for each trial

binomial random variable

the count of successes X in a binomial setting

• x = 0, 1, 2, …, n

calculating binomial probabilities

P(x = k) = nCk × p^k × q^n-k

• n = number of trials

• k = number of successes

• p = probability of success

• q = 1 - p = probability of failure

• nCk = n!/[k! × (n - k)!]

Formula for verifying binomial distribution

“y” is a binomial distribution with n = a and p(success) = b

• do not actually need to write out BINS, but must clarify the “N” and “S” part

convincing evidence

occurrence of an unlikely or likely event against the assumptions

• very small probability = against smth

• very large probability = for something

shapes of binomial distribution

• can be symmetric or skewed

  • when p = 0.5, the binomial distribution MUST be symmetric

  • when p ≠ 0.5. the binomial distribution MUST be skewed

    • p < 0.5 = right skewed

    • p > 0.5 = left skewed

mean of a binomial random variable

μ_x = n × p

• n = number of trials

• p = probability of success

standard deviation of binomial random variable

σ_x = (npq)^1/2

• n = number of trials

• p = probability of success

• q = probability of failure

formula for describing binomial distribution

The distribution of (variable name) is a binomial distribution (skewed towards__/symmetrically), with a peak around “a”. On average, the (_{variable name}) differs from the mean of “μ_x” (unit) by about σ_x (unit), when looking at “n” (unit).

10% condition

n < 0.10N

• when taking random sample of size “n” from a population size of “N”, we can use a binomial distribution to model the count of success in the sample is as long as the sample is less than 10% of the population

• approximately independent/binomial distribution