SU1 - Probability and Random Variables

Question 1

What is a probability space?

Answer 1

• sample space(Ω)
• sigma field(ℱ) field
• probability function(𝑷)

Question 2

What is a sample space?

Answer 2

Set of all the possible outcomes of a random experiment

Question 3

What is a sigma field?

Answer 3

set that contains all possible unions of all possible events arising from the experiment

Question 4

What are the three properties of a sigma-field?

Answer 4

• contains the sample space(Ω) itself
• contains each possible event and its complement
• all countable unions of events are contained in sigma field

Question 5

What are the three axioms of probability?

Answer 5

• the probability measure of this event must return a non-negative value
• probability function over the entire sample space is equal to 1
• for disjoint events, the probability of observing the union of these events is the sum of the probabilities of observing each of them

Question 6

What is a probability function?

Answer 6

provides the probabilities associated with the possible values of the random variable X

Question 7

What is a random variable?

Answer 7

They are numerical variables whose outcomes are not certain but are only known with a specific probability. A real-valued function that is defined on S

Question 8

What is a discrete random variable?

Answer 8

is a discrete random variable if X can take only a finite number k of different values or, at most, countably infinite number of values

Question 9

What is a Bernoulli Distribution?

Answer 9

a random experiment that has only two outcomes (usually called a “Success” or a “Failure”)

Question 10

What is a continuous random variable?

Answer 10

It can take on any real value along an interval, an infinite number of uncountable outcomes

Question 11

What are the two requirements of a pdf?

Answer 11

a) pdf must be non-zero (because as probabilities are non-zero, pdf naturally also non-zero)
b) the entire area under the density function must be equal to 1.

Question 12

What is a uniform distribution?

Answer 12

It has a constant pdf over the interval

see example in study guide

Question 13

What is cumulative distribution frequency?

Answer 13

It describes the probability of observing the random variable X up to a certain value, say x

Question 14

What are the 4 properties of cdf?

Answer 14

• A c.d.f F is a non-decreasing function of x.
  If x1 < x2, then P (X ≤ x1 ) ≤ P (X ≤ x2 ) i.e. F (x1) ≤ F (x2)
• P (X > c) = 1 -F (c)
• P (a < X ≤ b) = F (b) - F (a)
• P (a < X < b) = P (a ≤ X ≤ b) = P (a ≤ X < b) = P (a < X ≤ b)

Question 15

What is a quantile?

Answer 15

General expression for medians, quartiles, quintiles, percentiles, etc (median is 0.5, 0.25-quantile is the first quartile)

Question 16

How to test for independence?

Answer 16

P(B|A)=P(B)

Question 17

What is the probability density function?

Answer 17

Provides information on the likely outcomes of that random variable. Could be discrete/continuous

Question 18

What does the joint distribution tell you?

Answer 18

The dependence from one random variable to another

• joint probability function (if discrete)
• joint probability density function integrated over an interval (if continuous).

Question 19

What is a marginal distribution?

Answer 19

Distribution of one random variable.

• marginal probability function (discrete)
• marginal probability density function (continuous)

Question 20

What does the independence of two random variables mean?

Answer 20

Whatever the value of X is, will not affect the probability of Y

Question 21

What is the conditional density function?

Answer 21

Contains information about how a random variable X affects a random variable Y

Question 22

What is the conditional density?

Answer 22

The ratio of the joint density function and marginal density function