chapter 8.1

Random Variable

A variable that assigns a numerical value to each possible outcome of a random event.

Discrete Random Variable

A random variable with a finite or countable number of possible values.

Probability Mass Function (PMF)

Gives the probability of each possible value of a discrete random variable. p(x) = P(X=x).

Valid PMF Rules

Every probability must be between 0 and 1, and all probabilities must add up to 1.

Missing PMF Probability

Missing probability = 1 - sum of known probabilities.

Probability of Multiple Outcomes

Add the PMF probabilities for every outcome included in the event.

Cumulative Distribution Function (CDF)

Gives the probability that X is less than or equal to a specific value. F(x) = P(X<=x).

Expected Value

The probability-weighted average of a random variable. E(X) = sum of x times P(X=x).

Variance

Measures how spread out a random variable's values are around its expected value. Var(X) = E(X²) - [E(X)]².

Standard Deviation

Measures spread around the expected value. SD(X) = sqrt(Var(X)).

Joint PMF

Gives the probability that two discrete random variables take specific values at the same time. P(X=x,Y=y).

Marginal PMF

The probability distribution of one random variable found by adding joint probabilities across the other variable.

Conditional PMF

The probability distribution of one random variable given that another has a specific value. P(X=x|Y=y) = P(X=x,Y=y)/P(Y=y).

Conditional Probability Rule

The variable after "given that" is the denominator. Given Y, divide by the marginal probability of Y.

Conditional Expected Value

The average value of X when Y has a specific value. E(X|Y=y) = sum of x times P(X=x|Y=y).

Conditional Variance

Measures how spread out X is when Y has a specific value. Var(X|Y=y) = E(X²|Y=y) - [E(X|Y=y)]².

Joint CDF

The probability that two random variables are each less than or equal to specified values. F(x,y) = P(X<=x,Y<=y).

Expected Value of a Function

Multiply the function's value at each possible outcome by that outcome's probability, then add. E[g(X)] = sum of g(x) times P(X=x).

Linear Transformation

Multiplying a random variable by a constant and/or adding a constant. E(aX+b) = aE(X)+b.

Variance of a Linear Transformation

Multiplying X by a constant changes its variance by the square of that constant. Var(aX+b) = a²Var(X).

Standard Deviation of a Linear Transformation

Multiplying X by a constant changes its standard deviation by the absolute value of that constant. SD(aX+b) = |a|SD(X).

Expected Value of a Linear Combination

Multiply each random variable's expected value by its coefficient and add. E(aX+bY) = aE(X)+bE(Y).

Statistical Independence

Two random variables are independent if knowing one does not change the probability distribution of the other. P(X=x,Y=y) = P(X=x)P(Y=y).

Covariance

Measures how two random variables move together. Cov(X,Y) = E(XY)-E(X)E(Y).

Correlation

Measures the strength and direction of the linear relationship between two random variables. Corr(X,Y) = Cov(X,Y)/[SD(X)SD(Y)].

Linear Combination

A new random variable formed by multiplying variables by constants and adding them. V = k+aX+bY.

Variance of a Linear Combination

Var(k+aX+bY) = a²Var(X)+b²Var(Y)+2abCov(X,Y).

Sum of Independent Random Variables

Adding independent random variables together. For n independent variables with common mean mu and variance sigma², the sum has mean n*mu and variance n*sigma².

Average of Independent Random Variables

The sum of independent random variables divided by their number. For n independent variables with common mean mu and variance sigma², E(Xbar)=mu and Var(Xbar)=sigma²/n.