Week 3 - Probability and Random Variables

Question 1

Random Variable

Answer 1

• numerical summary of a random outcome.
• e.g. number of heads in two coin tosses in random variable.
• can take a range of values
• two types: discrete and continuous

Question 2

Discrete Variable

Answer 2

• Outcomes out countable
• e.g. 0,1,2,3,4
• each outcome has associated probability of occuring.

Question 3

Continious Variable

Answer 3

• outcomes not feasibly countable.
• can take any value in range of values.
• e.g. 1.5, 2.7898457, infinity.
• each outcome individually has 0 probability of occuring.
• Probabilities are associated with RANGE of outcomes.

Question 4

Sample space

Answer 4

All potential outcomes of random process.

Question 5

Probability Distribution of Random Variable

Answer 5

• List of all possibly values and probability that they will occur
• Probabilities add up to 1.
  e.g. pdf associated with X (where X is the number of heads in single coin toss) is written as f_x(X)

Question 6

Cumulative probability distribution

Answer 6

• Probability that the random variable is less than or equal to particular value
• Can be expressed using cumulative distribution function (cdf)
  e.g. P(X ≤ 0) = P(X = 0) = 0.5, P(X ≤ 1) = P(X = 0) + P(X = 1) = 0.5+ 0.5 = 1

Question 7

PDF for continuous variables

Answer 7

• The probability that X falls between

Question 8

CDF for continuous variables

Answer 8

• X ∈ [−∞, x] is the same as P[X ≤ x]
• Therefore, cdf is area under pdf up to x

Question 9

Expected Value

Answer 9

• Long run average value of the random value.

Question 10

Population Mean

Answer 10

The sum of all values in the population, denoted by the summation of X divided by the number of population values denoted by N.

Question 11

Mutually Exclusive and Exhaustive

Answer 11

Events cannot occur at the simultaneously , and one of them MUST occur.

Question 12

Probability useful rules

Answer 12

• P(A) = P(A ∩ B’) + P(A ∩ B)
• P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
• P(A ∪ B ∪ C) = P(A) + P(B) + P(C) − P(A ∩ B) − P(A ∩ C) − P(B ∩ C) + P(A ∩ B ∩ C)
• Bayes theorem:
  P(A|B) = P(A) x P(B|A)/P(B)
• if Independent: P(A|B) = P(A)