DTSA 5001

Question 1

factorial (!) operator

Answer 1

for any positive integer n:
n! = n * (n-1) * (n-2) * … * 2 * 1

Question 2

“n Choose r”

Answer 2

n! / (r! * (n - r)!)

Question 3

“n Permute r”

Answer 3

n! / (n - r)!

Question 4

variance and standard deviation

Answer 4

V = sigma^2

Question 5

P(A u B) = ?

Answer 5

P(A) + P(B) - P(A n B) = ?

Question 6

P(A u B u C) = ?

Answer 6

P(A) + P(B) + P(C) - (P(A n B) + P(B n C) + P(C n A)) + P(A n B n C) = ?

Question 7

A and B are independent if and only if…?

Answer 7

P(A n B) = P(A) * P(B)

Question 8

P(A’ n B’) = ?

Answer 8

1 - P(A u B) = ?

Question 9

P(A’ n B’ n C’) = ?

Answer 9

1 - P(A u B u C) = ?

Question 10

P(A’ u B’) = ?

Answer 10

1 - P(A n B) = ?

Question 11

P(A’ u B’ u C’) = ?

Answer 11

1 - P(A n B n C) = ?

Question 12

(A u B)’ = ?
Hint: De Morgan’s Law

Answer 12

(A’ n B’) = ?
Hint: De Morgan’s Law

Question 13

How many ways can a set S with n elements be arranged?

Answer 13

n!

Question 14

Multisets… Think of a question

Question 15

Probability of event E, P(E) = ?

Answer 15

Cardinality of Event over cardinality of Sample Space:
|E| / |S|

Question 16

Cardinality of sample when:
-Order matters (perm.)
-Sampled with replacement

Answer 16

n^k

Question 17

Cardinality of sample when:
-Order matters (perm.)
-Sampled without replacement

Answer 17

n! / (n - k)!

Question 18

Cardinality of sample when:
-Order doesn’t matter (comb)
-With replacement

Answer 18

(n + k - 1) choose k

Question 19

Cardinality of sample when:
-Order doesn’t matter (comb)
-Without replacement

Answer 19

n choose k

Question 20

Conditional Probability:
P(A|B) = ?

Answer 20

P(A n B) / P(B) = ?

Question 21

P(A|B)P(B) = ?

Answer 21

P(B|A)P(A) = ?

Question 22

P(A|B) = ?

Answer 22

P(B|A)P(A) / P(B)

Question 23

Law of Total Probability

Answer 23

P(A) = P(A|B)P(B) + P(A|B’)P(B’)

Question 24

Independent

Answer 24

One event doesn’t affect the probability of another;
P(A|B) = P(A)

Question 25

Binomial Distribution

Answer 25

P(X = x) = (n choose x) p^x * (1-p)^(n-x) where n = # trials, x = # true outcomes, p = single probability of true outcome - use when sampling with replacement

Question 26

Hypergeometric Distribution

Answer 26

P(X = x) = (d choose x) ((N - d) choose (n - r)) / (N choose n) where n = # trials, x = # true outcomes, N = pop size, d = # true outcomes in population - use when sampling without replacement

Question 27

Probability Combinatorics Step 1

Answer 27

Compute cardinality of sample space.

Question 28

Probability Combinatorics Step 2

Answer 28

Determine criteria for event or combination of events to be considered true: - consider permutations or combinations of sets or multisets. - If stuck try determining criteria for event(s) to be NOT true

Question 29

Probability Combinatorics Step 3

Answer 29

Compute cardinality of event space; i.e. how many ways are there to satisfy the criteria?

Question 30

Geometric Random Variable

Answer 30

An RV whose distribution represents the total number of independent Bernoulli trials (i) before the first success.

Question 31

Bernoulli Random Variable

Answer 31

An RV representing a trial which can only take on one of two values, success or failure, at a given individual probability (p).

Question 32

Binomial Random Variable

Answer 32

An RV whose distribution represents an unknown number of successes (i) in a fixed number of independent Bernoulli trials (n).

Question 33

Poisson Random Variable

Answer 33

An RV whose distribution represents the number of events (i) that happen within a fixed time interval (lambda).

Question 34

Uniform Random Variable

Answer 34

An RV whose distribution represents a constant probability within the interval [a, b].

Question 35

Exponential Random Variable

Answer 35

An RV whose distribution represents the time (lambda) between events.

Question 36

A measure of how strongly two RVs are related.

Answer 36

Covariance

Question 37

Covariance scaled to [-1,1]; often designated by the greek letter "rho."

Answer 37

Correlation coefficient

Question 38

Formula: sigma_X = ?

Answer 38

√(∑(x - µ_x)^2 * P(X = x))

Question 39

E(aX + bY + c) = ?

Answer 39

aE(X) + bE(Y) + c = ?

Question 40

Central limit theorem

Answer 40

The distribution of sample means approximates a normal distribution as the number of samples increases, regardless of the population's distribution.

Question 41

V(aX + bY + c) = ?

Answer 41

a^2 * V(X) + b^2 V(Y) + 2abCov(X,Y)

Question 42

Cov(XY) = ?

Answer 42

E(XY) - E(X)E(Y) = ?

Question 43

E(XY) = ?

Answer 43

∑(x * y * P(X = x, Y = y)) = ?

Question 44

The expected value of the sum of IRVs is...?

Answer 44

...the sum of the expected values.

Question 45

The variance of the sum of IRVs is...?

Answer 45

...the sum of the variances.

Question 46

Binomial Discrete PMF

Answer 46

P(X = i) = (n choose i) * p^i * (1 - p)^(n - i)

Question 47

Exponential PDF

Answer 47

f(x) = lambda * e^(-lambda * x)

Question 48

Exponential Expected Value

Answer 48

E(X) = 1/lambda

Question 49

Exponential Variance

Answer 49

V(X) = 1/lambda^2