Random Sample & estimation Flashcards
What is a random variable?
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
What is a support set?
The set of all possible values taken by a random variable
What are the properties of a random discrete variable?
- 0 <= pi <= 1 for i=1,2….k
p1+p2+….+pk=1
How do we calculate the Expected value for a discrete random variable?
E(x) = mean = x1p1+x2p2+……+xkpk
What do the variance and standard variation measure for discrete random variables?
The spread of values around the expected value. The variance is given by (x1-mean)^2p1+…..(xk-mean)^2pk
What are Bernoulli distributions?
An experiment that has 2 possible outcomes and one trial
What are the Bernoulli probabilities?
Value 1 = p
Value 0 = 1-p
What are the properties of the Bernoulli distribution?
E(x)= p Var(x) = p(1-p)
What is a binomial distribution?
Two outcomes, n trials
How do we describe the behavior of a continuous random variable?
Density function
What are the properties of a density function?
It only takes positive values and its area is equal to 1
What does the quantile (1-a) means for a continuous random variable?
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
What is the formula for expected value and variance of linear transformation?
E(y)=a+BE(x)
Var(y) = b^2Var(x)
What happens to the variance when we add a constant to the variable?
It remains constant
What is the formula for the standardization of a random variable?
(X-mean)/variance
What is the mean and variance of a standardized random variable?
0 and 1
In a normal distribution, for a given mean what happens when we increase the variance?
We have a flatter and wider bell-shaped curve
In a normal distribution, for a given mean what happens when we decrease the variance?
We have a taller and a narrower curve
What is the mean and variance of a normal distribution?
0 & 1
What is the probability equal to when falling 1 sd from the mean?
0.68
What is the probability equal to when falling 2 sd from the mean?
0.95
What is the probability equal to when falling 3 sd from the mean?
0.99
What is the expected value for two random vectors?
E(T) = a(meanx) x b(mean y)
Var (T) = a^2VARa + b^2VARb +2abCov(x,y)
What is the sum of two variables expected value and variance?
E(X+Y)=meanx+meany
Var(X+Y) = varx+vary
What is the subtraction of two variables expected value and variance?
E(X+Y)=meanx-meany
Var(X+Y) = varx+vary
What does the Central Limit Theorem states?
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
What are the properties of X for a large n based on the Central Limit Theorem?
It has mean m and variance equal to var/n
What is the mean and variance for Bernoulli variables based on the CLT?
mean = np Var = np(1-p)
What are the properties of X for Bernoulli variables based on the CLT?
mean = p Var = p(1-p)/n
