Lecture 1

Question 1

What is a distribution function?

Answer 1

This is a Cumulative distribution function, CDF.

The distribution function for a density function is a function that describes the probability that a random variable will take on a value less than or equal to a given number.

To find the distribution function, we need to integrate the density function over the given range.

Note. This

Question 2

How do you find the expectations of a density function?

Answer 2

E[X] = int_{lower limit}^{higher limit} xf(x) dx

Integrate this and we get the expectations.

Question 3

What is the formula for the variance?

Answer 3

VAR(X) = E(X^2) - E(X)^2

Question 4

How do you find the variance of a density function?

Answer 4

Use VAR(X) = E(X^2) - E(X)^2

First calsulate E(X) and square it. Then Calculate E(X^2) by

E[X^2] = int_{lower limit}^{higher limit} x^2f(x) dx

Then use the values in the the formula and calculate.

Question 5

How do you Calculate p(a≤ X ≤ b)

Answer 5

F(b) - F(a)

Question 6

How do you find p(X < a) ?

Answer 6

F(a)

Question 7

How do you find p(X > a) ?

Answer 7

1- F(a)

Question 8

What are the properties of a PDF?

Answer 8

Non-negative in the whole range.

The area is equal to one.

It is continuous.

Question 9

What is the formula for covariance?

Answer 9

cov[x,y] = E[xy] - E[x]E[y]

Question 10

Vad menas med density function och vad menas med distribution function?

Answer 10

density = PDF = f(x) = p(X = x)

distribution = CDF = F(x) = p(X ≤ x)

Question 11

What are the definitions of PDF vs CDF?

Answer 11

PDF = f(x) = A probability density function (pdf) tells us the probability that a random variable takes on a certain value. E.g., P(x = 1) : 1/6, using a dice example.

CDF = F(x) = A cumulative distribution function (cdf) tells us the probability that a random variable takes on a value less than or equal to x. E.g., P(x ≤ 3) : 3/6 using a dice example.

Question 12

WHat is the pdf of the uniform distribution?

Answer 12

f(x) = 1 /(b-a)

Question 13

What is the CDF of the unifrom distribution?

Answer 13

F(x) = (x-a)/(b-a) = (1/(b-a))(x-a)

Som i hennes exempel med U(0,2)

DÅ blir det

(1/(2-0))(x-0) = (1/2)x

Question 14

Difference between binomial and negative binomial

Answer 14

A binomial rv is the number of successes in a given number of trials.

A negative binomial rv is the number of trials needed for a given number of successes.

Question 15

What is the formula for the original CI?

Answer 15

Estimate + critical value x SE

Critical value is 1.96 if 95% conf

Question 16

What is formula for SE?

Answer 16

SE = sqrt (var/n)

Alternativly SE = Sd/sqrt(n)

Question 17

What is the formula for a t-test

Answer 17

t = (x - μ)/(sd/sqrt(n))

Where:
x = sample mean
μ= theoretical valu we will test against, should be = 0.
sd = standard diviation
N = sample size

Question 18

Write out the Likelihood ratio test

Answer 18

LR = 2(l(t^)-l(t))

Where l = log likelihood
t = thete = parameter of interest

We look at chi-2 distribution

Question 19

Write out the Wald test

Answer 19

(t^ - t)/sd(t^)

We look at chi-2 distribution.

Question 20

Write out score test

Answer 20

l’(θ_0)/squrt(l(θ_0)) ???