Chapter 1

Basic Concepts

1.1 Random Variables

• 1.1.1 Outcomes and Events

• (D,D) : The first and second pieces fall butter-side down.

• (D,U) : The first piece falls butter-side down and the second falls butter-side up.

• (U,D) : The first piece falls butter-side up and the second falls butter-side down.

• (U,U) : The first and second pieces fall butter-side up.

• {U,D}

• {(D,D), (U,D), (D,U), (U, U)}

• {U}, {D}, {U,D}, ∅

• {(D,D)}, {(D,U)}, {(U,D)}, {(U,U)}, {(D,D), (D,U)}, {(D,D), (U,D)}, {(D,D), (U,U)}, {(D,U), (U,D)}, {(D,U), (U,U)}, {(U,D), (U,U)}, {(D,U), (U,D), (U,U)}, {(D,D), (U,D), (U,U)}, {(D,D), (D,U), (U,U)}, {(D,D), (D,U), (U,D)}, {(D,D), (D,U), (U,D), (U,U)} ∅

• 1.1.2 Probability

• 1.1.3 Random Variables

• 1.1.4 Parameters

1.2 Binomial Distribution

1.3 Expected Values of Random Variables

Definition 15 (Expected Value)

• The expected value of a discrete random variable X is given as:

• Ex.

Definition 16 (Variance)

• The variance of a discrete randomvariable is given as:

• standard deviation

Definition 17 (Standard Error)

• The standard deviation of a parameter estimator is called the standard error of the estimator

Definition 18 (Two rules of expected values)

• E(X + Y ) = E(X) + E(Y )

•

• By Rule 1

• By Rule 2

1.4 Sequences of Random Variables

Definition 19 (Realization)

• The realization of a RV is the value it attains in an experiment.

Definition 20 (Relative Frequency Histogram)

Law of Large Numbers

Definition 21 (Sampling Distribution)

• A sampling distribution is the probability mass function of an estimator