Probability Distributions

Question 1

What is a random variable (RV)? and what are the two types or RV?

Answer 1

is a function that from the space sample Ω assigns a number x (real/natural/integer) to an outcome o. X(o)=x

Discrete random variable (DRV) which only takes discrete values and continuous random variable (CRV)

Question 2

What are the set of values SX of a RV X?

Answer 2

The set of values of the sample space Ω that are mapped under the RV X

e.g.

X= result of a coin toss, X(H)=1, X(T)=-1, SX={-1,1}

Question 3

What is the probability mass function (PMF) fx ?

Answer 3

Probability mass function (PMF) fx maps a discrete random variable X onto the probability measure P
fX(x)=P(X=x)

Question 4

What are the conditions of a PMF and PDF ?

Answer 4

• Positivity (non-negativity) f(x)≥0 for all x∈S
• Normalisation (total probability is ∑x∈S f(x) = 1 and
  ∫ from −∞ to ∞ fX (x)dx = 1

Question 5

What is a probability density function (PDF) fx?

Answer 5

The probability density function maps a continuous variable X to the probability that the random continuous variable X takes a value between x and x+dx.

P(a < x < b) = ∫ from a to b f X(x) dx

Question 6

What is the cumulative distribution fucntion (CDF) ?

Answer 6

A function FX that goes from [0,1] that maps a random variable to a probability that the random variable X takes a value smaller or equal to the given value x.

X (x) = P (X ≤ x)

Question 7

What is the relation between the cumulative probability distribution function (CPF) and the probability mass function (PMF) / probability density function (PDF)

Question 8

What is the definition of mean? is it always a possible outcome? write the matematical definition

Answer 8

no

Question 9

What is the definition of Variance Var(X) or σ^2? write the definition

Answer 9

Is the standard measure for the spread of the random variable X

Question 10

What is the definition of standard deviation? write the mathematical formula

Answer 10

σ = Var(X)

Question 11

What are the properties of the mean and of the variance?

Answer 11

• The mean is linear: E(aX + b) = aE(X) + b
• The variance is nonlinear V r(aX + b) = a^2var(X)