DTSA 5001 Flashcards
factorial (!) operator
for any positive integer n:
n! = n * (n-1) * (n-2) * … * 2 * 1
“n Choose r”
n! / (r! * (n - r)!)
“n Permute r”
n! / (n - r)!
variance and standard deviation
V = sigma^2
P(A u B) = ?
P(A) + P(B) - P(A n B) = ?
P(A u B u C) = ?
P(A) + P(B) + P(C) - (P(A n B) + P(B n C) + P(C n A)) + P(A n B n C) = ?
A and B are independent if and only if…?
P(A n B) = P(A) * P(B)
P(A’ n B’) = ?
1 - P(A u B) = ?
P(A’ n B’ n C’) = ?
1 - P(A u B u C) = ?
P(A’ u B’) = ?
1 - P(A n B) = ?
P(A’ u B’ u C’) = ?
1 - P(A n B n C) = ?
(A u B)’ = ?
Hint: De Morgan’s Law
(A’ n B’) = ?
Hint: De Morgan’s Law
How many ways can a set S with n elements be arranged?
n!
Multisets… Think of a question
Probability of event E, P(E) = ?
Cardinality of Event over cardinality of Sample Space:
|E| / |S|
Cardinality of sample when:
-Order matters (perm.)
-Sampled with replacement
n^k
Cardinality of sample when:
-Order matters (perm.)
-Sampled without replacement
n! / (n - k)!
Cardinality of sample when:
-Order doesn’t matter (comb)
-With replacement
(n + k - 1) choose k
Cardinality of sample when:
-Order doesn’t matter (comb)
-Without replacement
n choose k
Conditional Probability:
P(A|B) = ?
P(A n B) / P(B) = ?
P(A|B)P(B) = ?
P(B|A)P(A) = ?
P(A|B) = ?
P(B|A)P(A) / P(B)
Law of Total Probability
P(A) = P(A|B)P(B) + P(A|B’)P(B’)
Independent
One event doesn’t affect the probability of another;
P(A|B) = P(A)
Binomial Distribution
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
Hypergeometric Distribution
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
Probability Combinatorics Step 1
Compute cardinality of sample space.
Probability Combinatorics Step 2
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
Probability Combinatorics Step 3
Compute cardinality of event space; i.e. how many ways are there to satisfy the criteria?
Geometric Random Variable
An RV whose distribution represents the total number of independent Bernoulli trials (i) before the first success.
Bernoulli Random Variable
An RV representing a trial which can only take on one of two values, success or failure, at a given individual probability (p).
Binomial Random Variable
An RV whose distribution represents an unknown number of successes (i) in a fixed number of independent Bernoulli trials (n).
Poisson Random Variable
An RV whose distribution represents the number of events (i) that happen within a fixed time interval (lambda).
Uniform Random Variable
An RV whose distribution represents a constant probability within the interval [a, b].
Exponential Random Variable
An RV whose distribution represents the time (lambda) between events.
A measure of how strongly two RVs are related.
Covariance
Covariance scaled to [-1,1]; often designated by the greek letter “rho.”
Correlation coefficient
Formula: sigma_X = ?
√(∑(x - µ_x)^2 * P(X = x))
E(aX + bY + c) = ?
aE(X) + bE(Y) + c = ?
Central limit theorem
The distribution of sample means approximates a normal distribution as the number of samples increases, regardless of the population’s distribution.
V(aX + bY + c) = ?
a^2 * V(X) + b^2 V(Y) + 2abCov(X,Y)
Cov(XY) = ?
E(XY) - E(X)E(Y) = ?
E(XY) = ?
∑(x * y * P(X = x, Y = y)) = ?
The expected value of the sum of IRVs is…?
…the sum of the expected values.
The variance of the sum of IRVs is…?
…the sum of the variances.
Binomial Discrete PMF
P(X = i) = (n choose i) * p^i * (1 - p)^(n - i)
Exponential PDF
f(x) = lambda * e^(-lambda * x)
Exponential Expected Value
E(X) = 1/lambda
Exponential Variance
V(X) = 1/lambda^2
