STAT 101
Review for Exam 2
Exam 2 covers Chapter 14 - 17.
List of concepts:
1. Sample space
2. Events (subsets)
3. Union: A or B, write as A
S
B
4. Intersection: A and B, write as A
T
B
5. Two events are mutually exclusive (A
6. Two events are independent (P (A
T
T
B = Φ)
B) = P (A) ∗ P (B))
T
P (A B)
7. Conditional probability: P (B|A) =
P (A)
8. Probability model (aka Probability distribution): Use table, formula, histogram,
or relative frequency table to display all the values that the random variable can
take, and the corresponding probabilities.
9. Mean (expected value, expectation) of a discrete r.v. X: Multiply each value
of the r.v with its corresponding probability, then sum up over all the possible
values. Write as µ or E(X)
10. Variance: Write as σ 2 or V (X). Two formulas to compute.
11. Standard deviation: Write as σ. Square root of the variance.
List of rules:
1. Probability of the sample space is 1.
2. Complement: P (AC ) = 1 − P (A)
3. General addition rule (aka Inclusion-exclusion rule): P (A or B) = P (A)+P (B)−
P (A and B)
4. Know how to draw tree diagram.
5. Total Probability: To calculate the probability of an outcome on the 2nd branch
of the tree, sum up the paths lead to that outcome.
6. Bayes’ theorem: when there is a reversing conditional probability, use Bayes’s
theorem.
7. Read a story, know how to write down the probability model, compute the mean,
variance and standard deviation.
8. Linear combination of independent random variables:
Expecation is linear:
E(aX + bY ) = aE(X) + bE(Y ). e.g: E(2X − 3Y ) = 2E(X) − 3E(Y )
Variance is not linear: V (aX + bY ) = a2 V (X) + b2 V (Y ). e.g: V (2X − 3Y ) =
4V (X) + 9V (Y )
To compute standard deviation of a r.v, always compute the variance, then take
a square root.
9. Special distributions:
• Geometric distribution (model): X= the number of tosses until the 1st
heads. If P (H) = p, P (T ) = q,
P (X = k) = q k−1 p, k = 1, 2, 3, ...
• Binomial distribution: Y = the number of heads out of n tosses. If P (H) =
p, P (T ) = q,
P (Y = k) = nCkpk q n−k , k = 0, 1, 2, ..., n.
The mean E(Y ) = np. Variance V (Y ) = npq.
• Poisson Model: Counting number of occurrences pdf: P (X = k) =
0, 1, 2, ... The mean and variance are both λ
e−λ λk
,k=
k!