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MATH 526 – Exam I Priority Formula Sheet +

Exam-Style Problems

PART I: PRIORITY FORMULAS (Ordered by

Exam Likelihood)

Chapter 2: Probability

Section 2.3 – Conditional Probability

Theorem 2.3 (Conditional Probability)

P (A|B) = P^ ( PA (^ ∩B^ )B ), P (B) > 0

Multiplication Rule

P (A ∩ B) = P (A|B)P (B)

Independence P (A ∩ B) = P (A)P (B)

Section 2.4 – Law of Total Probability

Theorem 2. P (B) =

i

P (B|A i )P (A i )

Section 2.4 – Bayes’ Theorem

Theorem 2.5 (Bayes)

P (A i |B) = ∑P^ (B|A i )P^ (A i ) j P^ (B|A j^ )P^ (A j^ )

Chapter 5: Joint Distributions

Section 5.1 – Marginals

f X (x) =

y

f (x, y)

Section 5.2 – Covariance

Cov(X, Y ) = E[XY ] − E[X]E[Y ]

Correlation

ρ = Cov σ X (X, Y σ Y^ )

PART II: 5 EXAM-STYLE PROCEDURE

PROBLEMS

Problem 1 (Bayes)

A disease affects 4% of a population. If infected, the test is positive 95% of the time. If not infected, the test is positive 8% of the time. Find P (D|+).

Problem 2 (Binomial)

A component succeeds with probability 0.7 independently. In 12 trials:

1. Find P (X = 9).
2. Find P (X ≥ 10).
3. Compute E[X] and V ar(X).

Problem 3 (Normal)

Let X ∼ N (80, 16).

1. Find P (76 < X < 88).
2. Find c such that P (X < c) = 0. 90.

Problem 4 (Expected Value)

Given pmf:

x 1 2 3 P (X = x) 0. 2 0. 5 0. 3 Compute E[X] and V ar(X).

Problem 5 (Joint Distribution)

Joint pmf:

y = 0 y = 1 x = 0 0. 3 0. 2 x = 1 0. 1 0. 4

1. Find marginals.
2. Compute E[X], E[Y ], E[XY ].
3. Compute Cov(X, Y ).
4. Are X and Y independent?