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• Mathematical Economics and Econometrics (ECON1049) is the name of the the course
• The course is in Semester 2

Probability: Random variables and their properties

• Introduction; Examples and Properties of Random Variables will be covered
• The normal family of distributions plus Chebychev's Inequality will be covered
• Bivariate Random Variables - Correlation and Independence are included in the syllabus

Statistics: Estimation and hypothesis testing

• Samples and Sampling Distributions are included
• Estimation and Hypothesis Testing will be covered
• Applications of Tests and Confidence Intervals form part of the syllabus

Econometrics - The linear regression model

• Introduction to the Linear Model and Ordinary Least Squares (OLS) will be discussed
• Properties of the OLS estimators will be covered
• Extensions to the multivariable model, including specification will be studied

Properties of Random Variables

• An experiment where the results vary each time it is repeated is called a random experiment
• A set with all possible outcomes of a random experiment is a sample space
• An event is a subset of the sample space
• A random (or stochastic) variable (RV) is a variable whose value is determined via a random experiment
• Random variables are either discrete or continuous
• Continuous variables can take any value in a given range
• Discrete variables are constrained to take particular values

Discrete Random Variables

• A Bernoulli random variable is the simplest example of a discrete RV
• The probability the variable takes on the value one describes the behavior of a Bernoulli random variable
• If the coin is fair, then the probability that X equals one (head) is one-half; Pr[X = 1] = 1/2
• The probability that a Bernoulli RV equals 1 can be specified by 0; in principle, the 6 can be any number between 0 and 1.
• Pr[X = xk] = f (xk) k = 1, 2, . . . where X is a discrete random variable and x is a possible value
• The probability density function (PDF) is given by Pr[X = x] = f(x)
• For x = xk it becomes f (xk), while for other values of x, f(x) = 0
• For a discrete RV with sample space Ω the PDF f(x) is valid if i) f(x) ≥ 0, ii) ∑ f(x) = 1 for all values in the sample space

The coin is tossed twice with the sample space S = {HH, HT,TH,TT}

• X represents the number of heads that can come up
• With HH (i.e., 2 heads ), X = 2 while for TH(1 head ), X = 1
• The objective is to find its PDF

Binomial Distribution if

• Pr [X = x] = f (x) = x!( − x)!p² (1 − p)"−*; Ω = (0,1,2,...,n) and p > 0
• X ~ Bin [n, p] assigns probabilities to the number of successes in n independent trials with a prob of success p
• It is used in Decision/Game Theory and Economic phenomena where there is a discrete choice
• An important special case is the Bernoulli RV Bin [1, p]
• If each of Z1, …, Zn is B[1, p] (and independent of one another - independence is defined precisely later) then Z₁ + Z2 + .. + Zn is B[n, p]

Poisson Distribution if

• Pr [X = x] = f (x) = /x!; Ω = (0, 1, 2, ...., ∞) and μ > 0
• X ~ Po [μ] assigns probabilities to counts
• Examples: the number of people joining a queue in a given amount of time, or the number of people visiting a web page

Continuous Random Variables

• With a discrete RV, the PDF governs how probabilities are assigned to individual values
• With a continuous RV X, it gives the probabilities that X lies between two different values
• If X is a continuous RV on Ω = (-8,8), then the probability that X takes the value between a and b can be given by its PDF f(x) in the form of: Pr [a ≤ X ≤ b] = (integral from a to b) f (x)dx for -∞≤ a ≤ b ≤ ∞.
• Where the function f(x) has the properties: i) f (x) ≥ 0 ; ii) (integral from -∞ to ∞) f(x)dx = 1.
• probabilities for continuous RVs are only non-zero for intervals, not point outcomes
• Due to the number of possible values, a continuous random variable X can take on each value with probability zero
• Random variables that take on numerous values are best treated as continuous

Uniform Distribution if

• f (x) = 1/(b-a) ; Ω = [a, b]
• The notation is X ~ U[a, b]

Exponential Distribution if

• f (x) = (1/β)e-x/β; Ω = [0, ∞) and β > 0
• The notation is X ~ Exp [β]
• Such RVs are used to measure lifetimes, such as durations in unemployment in Labour Economics or survival times in Health Economics

Cumulative Distribution Function

• The cumulative distribution function (CDF) or briefly the distribution function, for a random variable X is defined by F(x) = Pr(X < x)
• x is any real number, i.e., -∞ < x < ∞

F(x) has the following properties

• F(x) is nondecreasing [i.e., F(x) < F(y) if x ≤ y ]
• limx→∞ F(x) = 0; limx→∞ F(x) = 1
• F(x) is continuous from the right [i.e., limp→0+ F(x + h) = F(x) for all x]

The CDF can be expressed as a function of the PDF, defined as

• F (x) = Pr [−∞ ≤ X ≤ x] = (integral from -∞ to x)) f (u) du (CONTINUOUS)
• F (x) = Pr [-∞ ≤ X ≤ x] = ∑f(u) (DISCRETE) where u<x
• For the continuous case, interval probabilities are Pr [a ≤ X ≤ b] = (integral from a to b) (f (x)dx = (integral from -∞ to b) (f(x)dx - (integral from -∞ to a) (f (x) dx = F (b) - F (a),

Moments

• Main characteristics of a PDF (and the RV it assigns probabilities for) can be summarized by moments
• The first moment is known as the mean/expected value
• The second moment relates to the spread or variance of the RV
• For a RV X its mean is a measure of the average value/location
• Variance is a measure of dispersion around that mean
• The mean (or variance) are not RVs, but are fixed numbers that are functions of the PDF

Expected Value

• The expected value or mean of a RV X, E [X] (or μ), is defined as: E [X] = (integral from −∞ to ∞) x f (x)dx (CONTINUOUS); E [X] = (summation from x=−∞ to ∞) xf (x) (DISCRETE)
• If g (X) is a continuous function of X then we define: E [g (X)] = (integral from −∞ to ∞) g(x) f (x) dx or E [g(X)] = (summation from x=−∞ to ∞) g (x) f (x)
• If a and b are constant: E [aX + b] = (integral from minus infinity to infinity) (ax + b) f (x) dx = a * (integral from minus infinity to infinity) x f (x) dx + b *(integral from minus infinity to infinity) f (x) dx= aE [X] + b, which is also true if X is a discrete RV
• If the continuous RV X has a PDF which is symmetric around a point a, then E [X] = a

Variance

• The variance, denoted V [X] (or σ²X), is defined as V [X] = E [(X – E [X])²]
• The variance is a nonnegative number
• The positive square root of the variance is called the standard deviation and is given by σX = √Var[X] = E [(X – E[X])²]
• Note that it is possible to define the variance as E [(X - E [X])²] = E [(X² – 2E[X]X + E[X]²)] = E [X²] – E[2E[X]X] + E[E[X]2] =E [X2] – 2E[X]E[X] + E[X]² = E [X2] – E[X]^2
• If a and b are any constants, V(aX + b) = a² V(X)

Moments

• The rth moment of X about the origin, also called the rth raw moment (or just rth moment), is defined as μ'r = Ε[Χ^r]
• The rth moment of a random variable X about the mean µ, also called the rth central moment, is defined as μr = Ε [(X − μ)^r]
• Expected value is the first moment while variance is the second central moment
• Higher order moments can be calculated to define other features of the distribution of a RV
• The third moment of the standardized version of a RV is viewed as a measure of skewness
• The fourth moment of the standardized version of a RV is called a measure of kurtosis

Moment Generating Functions

• Moments of even simple RVs can be difficult to calculate, especially the higher order moments
• The mean of RV e^λy, which is a function of RV Y (with a PDF f(y)) = E (e^λy) = (integral from -∞ to ∞) e^λy f(y)dy, and this is a function of λ which we denotes by Mγ(λ
• Note the exponential function definition, e^x (λ)= (summation from i=0 to ∞)((λx)^i)/i! = 1+ λy + (λy)^2/2!+..+ ((λy)^i)/i!..
• My(λ) = integral from -∞ to ∞)e^λy f(y)dy = (integral from -∞ to ∞) (1+ λy + (λy)^2/2!+..+ ((λy)^i)/i!.. )f(y)dy = 1+ λE[Y]+ (λ^2/(2!))E[Y^2]+..+(λ^i/(i!))*E[Y^i]....
• So that if we take the ith derivative (so that every term to the left of E[Yi] disappears) and evaluate at λ = 0 (so that every term to the right of E[Yi] evaluates to zero) we get (d^i My(λ))/(dλ) = E[Y^i] given λ = 0
• The moment generating function (MGF) of Y is My (λ)

The MGF satisfies two very important properties:

• (i) Let Y be a random variable having MGF My(λ), then for g(Y) = aY + b, Mg(Y) (λ) = ebλ My (αλ)
• (ii) If X and Y are independent and Z = X + Y, then Mz(λ) = Μx(λ)Μγ (λ), i.e. if we add RVs together the resulting MGF is the product of the MGFs of the individual RVs we are adding
• The MGF is a fundamental characteristic of an RV
• There is a one-to-one correspondence between the PDF and MGF

The Change of Variable Formula

• It is useful in properties of functions of RVs
• Suppose that a RV X has PDF f (x) for a ≤ X ≤ b and distribution function F (x) and let Y = h (X) where h (.) is a monotonic (whether increasing or decreasing) function
• The density and distribution functions of Y, g (y) and G (y) then the Change of Variable formula states: g (y) = f (x) * abs(dx/dy)
• Assume first that h(.) is increasing (dh(x)/dx > 0)then h (a) < Y <h (b)
• The dist formula is: G (y) =Pr [Y ≤ y] = Pr [h (X) ≤ y] = Pr [X ≤ h¯¹ (y)] = F (h-1 (y)) = F(x), where x = h¯¹ (y)
• Get the density, differentiate, g (y) d/dy G (y) = d/dx F (x)* dx/dy = f (x) dy
• The density is then g (y) = d/dy G (y) = d/dx (1 – F (x))* dx/dy =− f (x) dy dx /dy
• Brings both results together is: g (y) = f (x)|dx/dy|

The Normal Family of RVs

• A continuous RV X has a normal (or Gaussian) distribution if its PDF is the form f(x) = 1/(sqrt(2πσ² )) * exp{- 1/(202) * (x- μ)^2}; -∞ ≤ X ≤ 8.
• Then this RV X is a normal random variable, denoted by X ~ Ν(μ,σ²)
• This distribution is completely characterized by the two above parameters which can meet the following; Ε[X] = (integral(-inf to inf)) xf(x)dx = μ, V[X] = (integral(-inf to inf))(x-μ)^2 f(x)dx = σ²

CDF is often written as

• Pr[X ≤ x] = Φ(x) = (integral from -∞ to x) 1/(sqrt(2π ))* e^(-u^2)/2 *du
• The density function for the normal distribution is bell shaped and symmetric about μ. For Standard normal RV has mean 0 and variance 1, often denoted by Z ~ N (0,1).

Main properties:

• Pr[IX-μI > σ] = 0.3173, Pr[IX-μI > 2σ] = 0.0455 etc.
• For Standard normal RV Z for Useful expression any C >0: Pr[Z > c] = Pr[Z > c] + Pr[Z Pr =2P[Z>c] = 2[1 − Ф(c)]
• If X distributed: If X ~ Ν (μ, σ²) then aX + b is also normally distributed ~ N (αμ + b, a²σ²)
• We calculated E [aX + b] = aE [X] + b and V [aX + b] = a²V [X], however the result here is much stronger in that says linear function a normal RV Normal
• IF X1 + X2 normally distributed: We can show if X₁ ~ N (μ₁, σ₁) also that is X2 ~ Ν (μ2, σ2,) then X1 = X2 are also independent, Y=aX1 +aX2~ N(αιμι + a2μ2, ασ + ασξ), and linear independent, normal RVs normal RV.
• IF- X ~ Ν (μ, σ2,) Then If X ~N(μ, σ²) = Pr[- a < X < b] = Pr[- a < X < b] / sigm - μ/sigma.
• The formula for the change in probability when you apply property 3 in the eq. is that Z= X - μ /sigma - ~N (0,1)
• That can also be easilly converted also into 0.126
• In summary: The N(0,1) from Table of Probability

The Chi-Square Distribution

• Let Xi ~ N (0, 1) for =1, ...n Let Xi for independent.
• Equation is Q(n) = n (sigma) xi ^2

Then formula is used with Degrees of

• The Chi Square Distribution with writing: Q(n) ~ X(n)

Properties of this Distribution:

• Is has to be normally symmetric and that: X (sigma) with PDFS
• Show that E (Q(n). - n
• Independent fordeducement that square RVS
• Sque tables required different volume

The Student t Distribution

• LetZ Normally distribution. 0.1 with let Q distribution X(2) where Q independent then

Equation

• T=/ Square n = Qn
• Student With Distribution We T = (10)

Properites

• Dersity that has to be has tails
• Distribution to be T(0.1)

Student

• The graph of the equation freedom is showb here for the values freedom is shown
• 0.126 shadeed are is graph

F the the Distribution

• Square

And with distribution is then equation: Q1/n1 = F= Q2/n2

• Has with normally Properties
• The equation has with distribution is F

Chebyshev's Inequality

• For any RV X w/ PDF f(x), standard mean and variance , the equation will true. Where there all bounds
• The formula show equation show true that where or will suppose
• Chebysheve suppose the is show equation we will be be useful data this section.

Bivariate RVs

• Let Y is equal where formula is

Equal the is then

• Multiple integrals can easily

Marginal PDS

Are The formula equation with the follows.

• (Xx = infinity)

Stastical Indepencdence

• The independent where and can

Conditional

formula has to be

Covarirance and Correclation

• We equation between the can show that the indepdent