Random Variables

random variable

a function that assigns a numerical value to each point in the sample space

describes the uncertain outcomes of a random process

denoted by x

probability distribution

lists the possible outcome (x) for a random variable (X) + their associated probabilities

discrete

a random variable that takes on one of a list of possible values (counts)

continuous

a random variable that takes on any value in an interval

mean

weighted sum of possible values with probabilities as weights

denoted by μ

also referred to as the expected value of X or E(X)

standard deviation

σ = SD(X) = root(var(X))

add a constant to X → X ± c

E(X ± c) = E(X) ± c

Var(X ± c) = Var(x)

SD(X ± c) = SD(x)

multiply x by a constant → cX

E(cX) = cE(X)

SD(cX) = |c|SD(X)

Var(cX) = c²Var(X)

addition and multiplication rules

E(a+bX) = a + bE(X)

SD(a + bX) = |b|SD(X)

Var(a + bX) = b²Var(X)

joint probabilities

NOT independent events

need joint probability distribution that gives probabilities for events of the form (X=x, Y=y)

independence relationship

two random variables are independent if + only if the joint probability distribution is the product of the marginal distributions → P(x,y) = P(x)P(y) for all x,y

expected value relationship

the expected value of a product of independent random variables is the product of their expected values → E(xy) = E(x)E(y)

variance relationship

the variance of the sum of independent random variables is teh sum of their expected values → Var(x+y) = Var(x) + Var(y)

*not necessarily the sum of the variances

expected value relationship

the expected value of a sum of random variables is the sum of their expected values → E(x+y) = E(x)+E(y)

covariance

the covariance between random variables is the expected value of the product of deviators from the means → Cov(x,y) = E((x-μ_x)(y-μ_y)

correlation

the correlation between two random variables is the covariance divided by the product of standard deviations

Corr(x,y) = (Cov(x,y))/((σx)(σy))

standardized measure of the association between two random variables

denoted by ρ (rho)

always between -1 and 1

bernoulli

0 or 1 (failure or success)

E(B) = p // Var(B) = p(1-p)

fixed probability of success (p)

independence

binomial

y, the sum of iid bernoulli random variables, is a binomial random variable; fixed number of trials (n), fixed number of successes (p) (independence)

y= number of successes in n bernoulli trials (each trial with probability of success p)

P(Y=y) = (n!)/((n-y)!y!) * p^y(1-p)^n-y

E(Y)=np // Var(Y) = np(1-p)

poisson

number of events in interval

describes the number of events determined by a random process during an interval of time or space (couonts)

E(X) = λ // Var(X) = λ

P(X=x) = e^-λ*(λ^x/x!)

independently + identically distributed (iid)

random variables that are independent of each other + share a common probability distribution are said to be independent + identically distributed

if n random variables are iid with mean μ_x and standard deviation σ_x

continuous random variables

counts don’t work for everything

prices, costs, revenue, etc. ($), percent change, other measurement data

uniform random variables

lower bound (a), upper bound (b)

equally likely to be any value in between a + b

E(X) = (b-a)/2

Var(X) = (b-a)²/12

normal random variables

visualizing data: continuous range of values (histrogram, no skew)

defined by parameters μ + σ² (smaller σ² = narrow)

is continuous + can assume any value in an interval

standard normal

Normal (μ=0, σ²=1) = probability density function (pdf)

values more likely to be closer to mean

entire area under the curve = 1

z-scores

for any value from the given normal distribution, we can convert it to a value from the standard normal distribution

z= (x-μ)/σ

multimodaility

more than one mode suggests data comes from distinct groups

skewness=lack of symmetry

outliers = unusual extreme values

parameter

a characteristic of the population

statistic

an oberseved characteristic of a sample

central limit theorem (CLT)

tells us that if n is sufficiently large, the distribution of sample means is normally distributed