Topic 3: Special Distributions

Question 1

What is a bernoulli distribution (2,1)

Answer 1

-A bernoulli distribution is one with 2 outcomes, a success’ and a ‘failure’
-x = 0 (failure), with probability 1-p and x = 1 (success), with probability p

-This is called the w variable in econometrics, looking on whether you have/not have a characteristic

Question 2

What is the expected value/variance of a bernoulli trial (2)

Answer 2

E(X) = p
V(x) = p(1-p)

Question 3

What is a binomial distribution (2)

Answer 3

-A binomial distribution has n independent trials, with X = the number of successes
-The probability of a specific outcome X=r = (nCr)p^r(1-p)^n-r

Question 4

What are the expected value and variance for a binomial distribution (2)

Answer 4

-E(X) = np
-V(X) = np(1-p)

Question 5

What is a poisson distribution (3,1)

Answer 5

-A poisson distribution looks at the number of occurances of an event happening over a period of time T, divide into n unit time intervals (n is large)
-The probability of an even in a unit time interval is p (p is small), and events are independent occurences
-λ = the mean rate of occurance, the arrival rate

-Px(X) = e^-λλ^x/x! = the probability of x = 0, 1, 2, ….

Question 6

What is the expected value and variance for a poisson distribution (2)

Answer 6

-E(X) = λ
-V(X) = λ

Question 7

What is a uniform probability density function (2)

Answer 7

-A uniform probability density function is one where all points in (a,b) have a equal likelihood of occuring
-Fx(x) = 1/(b-a)

Question 8

What is the expected value and variance of a uniform probability density function (2)

Answer 8

-E(X) = a+b/2 (same as median, mode due to symmetry) (integrate x/(b-a) with b and a)
-V(X) = (b-a)²/12

Question 9

What is the density function of a normal distribution (1)

Answer 9

-(2πσ)^-1/2 e^{(-(x-μ)2/2σ2)}

Question 10

What is the expected value and and variance of a normal distribution (2,1)

Answer 10

-E(X) = μ
-V(X) = σ²

-X~N(μ, σ²)

Question 11

What is the standard normal distribution (1)

Answer 11

• Z~N(0, 1)

Question 12

What is the formula to convert a normal into a standard normal (1)

Answer 12

• Z = (X-μ)/σ

Question 13

How do you combine different normal distributions together (3,1)

Answer 13

-Let X₁ ~ N(μ, σ²) and X₂ ~ N(μ, σ²)
-To work out the mean, you add the coefficients then times that by the mean (X₁ coefficient = 1, -X₂ coefficient = -1 etc)
-To work out the standard deviation you square the coefficients, then add those together and times that by the variance (both X₁ and -X₂ have coefficient 1)

-Since X₁ and X₂ are independent drawings from distributions, the coefficient is 0 and V(X₁ + X₂) = V(X₁) + V(X₂)

Question 14

What does a normal distribution look like (3)

Answer 14

-Have an x axis with infinity to the right, and -infinity to the left
-The normal distribution is a symmetric bell shaped curve, centered around the origin
-The area under the curve is 1, so this is a valid PDF

Question 15

How do we define the pdf for a Chi-squared distribution + one with 1 DoF (3,1)

Answer 15

-f_X(x) = (1/Γ(v/2))(1/2)^(v/2)x^(v/2)-1e^-x/2, x > 0
-Γ(v) = vΓ(v-1) and Γ(1/2) = √π
-This is a valid probability density function and denoted as χ²_v, where v are the degrees of freedom

-Pdf of a χ²₁ = (x^-1/2)(e^-x/2)/(√2π)

Question 16

What is the link between a standard normal distribution and a chi-squared distribution (1)

Answer 16

-N(0, 1)² = χ²₁

Question 17

What is the expected value and variance of a chi-squared distribution (2)

Answer 17

-E(X) = v
-V(X) = 2v

Question 18

How do you draw a chi-squared distribution (3)

Answer 18

-Have chi square on the x axis, and density on the y
-The distribution will likely slope upwards first, hit a max point then start sloping down, eventually going from concave to convex
-The diagram will change depending on the number of degrees of freedom

Question 19

How do you add up independent chi-squared distributions (2)

Answer 19

-You sum the degrees of freedom
-χ²₁ + χ²₁ = χ²₂

Question 20

What is an F distribution (2,1)

Answer 20

-An F distribution is formed as the ratio of 2 independent chi-squared distributions
-(χ²_m/m) / (χ²_n/n) ~ F_m, _n

-If n tends to infinity, The F distribution can be approximated as χ²_m/m

Question 21

What is the PDF for an F distribution (1)

Answer 21

-f_X(x) = (Γ[(m+n)/2])/(Γ(m/2)Γ(n/2)) * (m/n)^m/2 * (x^(m-2)/2)/([1+(m/n)x]^(m+n)/2), x>0

Question 22

What is the expected value and variance for an F distribution (2)

Answer 22

-E(X) = n/n-2
-V(X) = (2n²(m+n-2))/(m(n-2)²(n-4))

Question 23

What does an F distribution look like (3)

Answer 23

-Have F on the x axis, and density on the Y axis
-The curve will start at 0, concavely shoot up then start falling, and turn convex eventually
-The shape will depend on the degrees of freedom of both the numerator and denominator

Question 24

What is a T distribution (2,1)

Answer 24

-A t-distribution is formed as the ratio of a standard normal distribution to a chi-square distribution
-t_n ~ (N(0,1))/√((χ²_n)/N)

-As n tends to infinity, t_n ~ N(0, 1)

Question 25

What is the PDF for a T-distribution (1)

Answer 25

-f_X(x) = (Γ[(n+1)/2])/(Γ(n/2)) * (1/√nπ) * (1/(1+x<up>2</sup>/n)^(n+1)/2)</up>

Question 26

What is the mean and variance of a t distribution (2)

Answer 26

-E(x) = 0
-V(X) = n/n-2

Question 27

What does a t distribution look like (3)

Answer 27

-Have T on the x axis and density on the y axis
-Similar to a normal distribution, a T distribution is a bell shaped symmetrical curve based around the origin
-However, how tall/flat it is depends on the DoF (more DoF = taller, infinite = standard normal)

Question 28

What is the relationship between an F and T distribution (1)

Answer 28

-(t_k)² = F_{1, k}

Question 29

How do you take a poisson approximation to a normal distribution (1)

Answer 29

-X~N(μ, σ²) ≈ X ~ P(μσ)