L3 - Random Variables - Discrete

Question 1

What is a random variable?

Answer 1

A random variable is the result of a chance situation, so its value depends on chance: we have a range of possible outcomes and a set of
probabilities associated to them. For example:
- The number of customers that will enter a shop in the next hour;
- The number of children in a family;
- The height of people;
- The number I get when throwing a die

The numerical outcome of a random variable can be either discrete or
continuous

Question 2

How is a Discrete Random Variable defined?

Answer 2

• Discrete r.v.
• only finite values within a range, usually integers
  (ex. the die). The sample space is defined over a finite number of
  outcomes.

Question 3

How is a Continuous Random Variable defined?

Answer 3

• Continuous r.v
• any value within a range, so the number of values is virtually infinite. The sample space has an infinite number of outcomes.
• Height or temperature which can be measured to any degree of accuracy;
  but also the number of unemployed people in a country, or the expenditure
  of a family in a year
  . - Although finite and not divisible, the number of values the variable can take on is so large that for all practical purposes it can be
  considered continuous.

Question 4

How is a Probability distribution defined?

Answer 4

a list of all the possible outcomes of a discrete r.v. and their associated probabilities.

• using X for the outcomes and P(x) for their respective probabilities
• ΣP(x) = 1

Question 5

For Discrete random variable what is its probability when its represented on a graph?

Answer 5

• the probability of an outcome is exactly identified and can be read on the vertical axis

Question 6

What is Cumulative Probability?

Answer 6

• The cumulative probability is the probability that the variable lies within a certain range or is larger or smaller than a certain values.
• Cumulative probabilities are calculated as sums:
• F(x) = P(X ≥ x{a}) = Σ_(X ≥ x{a}) (P(x)) or 1- P( x < x{a})
• F(x) = P(X ≤ x{a}) = Σ_(X ≤ x{a}) (P(x))
• F(x) = P(X{a} ≤ X ≤ X{b}) = Σ_X{a}^X{b} (P(x))

Question 7

How do you find the Mean of a Random Variable?

Answer 7

• The mean of a r.v. X is defined as the “expected value of X”
• E(X) = μ = ΣxP(x)
• In the sample you had x(bar) = Σx/n = Σx(1/n)
• concept is the same since the same P(x) = 1/n
• μ and x(bar) are two different things; μ is a parameter of the population,
  so it refers to the “true” distribution of X. The sample mean is an
  estimator, a statistic from a sample of the population (which is often all one has). Different samples will have different estimates.
• The larger the sample is the closer x(bar) will be to μ. This is because the sample mean is a consistent estimator of the population mean.

Question 8

How do you calculate the Variance of Random Variables?

Answer 8

• The variance of a r.v. is defined as:
  σ^2 = E[(X- μ )^2] = E(X^2) - μ^2

And is computed as:
σ^2 = [Σx^2P(x)] - μ^2

The standard deviation is defined as σ = sqrt(σ^2)

• Some variables and their distributions are often defined simply by their
  mean and variance parameters; in that case we summarize the information as:
  X ~ (μ,σ^2 )