Probability and Random Variables

Question 1

P(A) and P(B) are mutually exclusive if…?

Answer 1

P(A∩B) =0
“A intersect B” = 0

Question 2

P(A) and P(B) are independent if…?

Answer 2

P(A∩B) = P(A) * P(B)

Question 3

Probability Rule #3

Answer 3

P(A∪B) = P(A) + P(B) - P(A∩B)

Question 4

Order matters (permutation)

Answer 4

nPk = n! / (n-k)!

Question 5

Order doesn’t matter (combination)

Answer 5

nCk = n! / k!*(n-k)!

Question 6

Equally likely outcome

Answer 6

(# of outcomes in A) / (# of outcomes in S)

Question 7

Equally Likely Outcomes

Answer 7

(# of outcomes in A) / (# of outcomes in S)

Question 8

What is a random variable?

Answer 8

A random variable is the number of possible outcomes of some event.
Random variables can be continuous (inclusive of decimal values) or discreet (whole numbers only)

Question 9

What is a probability mass function?

Answer 9

A probability mass function or PMF, puts each RV into a table format with the probabilities of each outcome below the RV
Tossing a coin 2x, and landing on heads
RV is possible times you could land on heads
RV = x | 0 1 2
P(x) | 1/4 1/2 1/4
RULE 1: for any x, P(x) is greater than or equal to 0
RULE 2: ΣP(x) = 1

Question 10

What is a cumulative distribution function?

Answer 10

A cumulative distribution function, or CDF is basically a PMF except that you add the probabilities of each RV up to and including that RV
To obtain the CDF, integrate the function of the PMF
CDF may include a column or row of x<1 which in the case of tossing a coin twice is 0
Also, in our example, for x>2, no matter the value is 1
RV = x | 0 1 2
P(x) | 1/4 3/4 1

Question 11

Expected Value

Answer 11

Expected value or the mean (long term average)
μ(x) = E(x) = Σ (x * P(x))

Question 12

Variance

Answer 12

σ^2 = E(x^2) - [E(x)]^2
NOTE: for the E(x^2) term, square the RV (x-value) not the probability. Multiply the x^2 value and each corresponding probability, add all terms for E(x^2) value

Question 13

Standard Deviation

Answer 13

σ

Question 14

Finding probability of a continuous random variable (RV)

Answer 14

integrate the f(x) between values of x (a & b)

Question 15

Finding expected value of a continuous random variable

Answer 15

include one x value (RV) into the integration term so the integration will be over x*f(x)

Question 16

E(x^2)

Answer 16

E(x^2) = include x^2 in the integration from a to b

Question 17

variance

Answer 17

σ^2 = E(x^2) - [E(x)]^2

Question 18

Counting

Answer 18

Just multiply the number of each option in each place (NO FACTORIALS!!!)

Question 19

And (intersect)

Answer 19

MULTIPLY

Question 20

OR (union)

Answer 20

ADD

Question 21

A system in series

Answer 21

in series: P(A∩B) = P(A) * P(B)

Question 22

A system in parallel

Answer 22

In parallel: P(A∪B) = P(A) + P(B) - P(A∩B)