Random Sample & estimation

Question 1

What is a random variable?

Answer 1

Variable –> before drawing the unit we only know that different values can be observed
Random –> before drawing we don’t know the values we are going to observe
.:. It’s a numerical measurement of the outcome of an experiment

Question 2

What is a support set?

Answer 2

The set of all possible values taken by a random variable

Question 3

What are the properties of a random discrete variable?

Answer 3

• 0 <= pi <= 1 for i=1,2….k

p1+p2+….+pk=1

Question 4

How do we calculate the Expected value for a discrete random variable?

Answer 4

E(x) = mean = x1p1+x2p2+……+xkpk

Question 5

What do the variance and standard variation measure for discrete random variables?

Answer 5

The spread of values around the expected value. The variance is given by (x1-mean)^2p1+…..(xk-mean)^2pk

Question 6

What are Bernoulli distributions?

Answer 6

An experiment that has 2 possible outcomes and one trial

Question 7

What are the Bernoulli probabilities?

Answer 7

Value 1 = p

Value 0 = 1-p

Question 8

What are the properties of the Bernoulli distribution?

Answer 8

E(x)= p
Var(x) = p(1-p)

Question 9

What is a binomial distribution?

Answer 9

Two outcomes, n trials

Question 10

How do we describe the behavior of a continuous random variable?

Answer 10

Density function

Question 11

What are the properties of a density function?

Answer 11

It only takes positive values and its area is equal to 1

Question 12

What does the quantile (1-a) means for a continuous random variable?

Answer 12

There is a probability of (1-a) of observing a below that quantile and a probability of a of observing a value above that value. It refers to the area that is left above the value

Question 13

What is the formula for expected value and variance of linear transformation?

Answer 13

E(y)=a+BE(x)

Var(y) = b^2Var(x)

Question 14

What happens to the variance when we add a constant to the variable?

Answer 14

It remains constant

Question 15

What is the formula for the standardization of a random variable?

Answer 15

(X-mean)/variance

Question 16

What is the mean and variance of a standardized random variable?

Answer 16

0 and 1

Question 17

In a normal distribution, for a given mean what happens when we increase the variance?

Answer 17

We have a flatter and wider bell-shaped curve

Question 18

In a normal distribution, for a given mean what happens when we decrease the variance?

Answer 18

We have a taller and a narrower curve

Question 19

What is the mean and variance of a normal distribution?

Answer 19

0 & 1

Question 20

What is the probability equal to when falling 1 sd from the mean?

Answer 20

0.68

Question 21

What is the probability equal to when falling 2 sd from the mean?

Answer 21

0.95

Question 22

What is the probability equal to when falling 3 sd from the mean?

Answer 22

0.99

Question 23

What is the expected value for two random vectors?

Answer 23

E(T) = a(meanx) x b(mean y)

Var (T) = a^2VARa + b^2VARb +2abCov(x,y)

Question 24

What is the sum of two variables expected value and variance?

Answer 24

E(X+Y)=meanx+meany

Var(X+Y) = varx+vary

Question 25

What is the subtraction of two variables expected value and variance?

Answer 25

E(X+Y)=meanx-meany

Var(X+Y) = varx+vary

Question 26

What does the Central Limit Theorem states?

Answer 26

It says that the sum of n variables independent and identically distributed with the same mean and variance has distribution that for a large n cam be approximated by a normal distribution with mean nmean and variance nvar

Question 27

What are the properties of X for a large n based on the Central Limit Theorem?

Answer 27

It has mean m and variance equal to var/n

Question 28

What is the mean and variance for Bernoulli variables based on the CLT?

Answer 28

mean = np 
Var = np(1-p)

Question 29

What are the properties of X for Bernoulli variables based on the CLT?

Answer 29

mean = p
Var = p(1-p)/n