Download 1. Discrete random variables. Let X be a discrete random variable

1. Discrete random variables. Let X be a discrete random variable with possible values
1, 2 and 3 which probabilities of these values are: P (X = 1) = 0.5, P (X = 2) = 0.2
and P (X = 3) = 0.3). Answer the following questions:
• What is the value of µX = E(X)?
• Explain in words what µX represents.
2 = V ar(X)?
• What is the value of σX
2 represents.
• Explain in wods what σX
• What is the value of F (2) = P (X ≤ 2)?
• What is the value of P (X > −2)?
Define a new random variable Y = 2X 2 + 3. Answer the following questions:
• What are the possible values of Y and their probabilities?
• What is the value of µY = E(Y )?
• What is the value of σY2 = V ar(Y )?
• What are the values of Cov(X, Y ) and ρXY = corr(X, Y )?
• What does ρXY represent?
• Plot FY for all the possible values of Y .
2. Continuous random variables. Let Z be a continuous random variable with a standard distribution N (0, 1). Answer the following questions:
• What are the possible values of Z?
• What is the value of µZ = E(X)? What is i
• Explain in words what µZ represents.
• What is the value of σZ2 = V ar(X)?
• Explain in wods what σZ2 represents.
• What is the expression of density function?
• What is the expression of the distribution function?
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• What is the expression of the mean?
• What is the expression of the variance?
Define a new random variable Y = 3Z + 2. Answer the following questions:
• What are the possible values of Y ?
• What is the expression of the density function (fY ) of Y ?
• What is the expression of the distribution function (FY ) of Y ?
• What is the value of µY = E(Y )?
• What is the value of σY2 = V ar(Y )?
• What are the values of Cov(Z, Y ) and ρZY = corr(Z, Y )?
• Plot FY for all the possible values of Y .
3. Random variables and R. Intall R and RStudio in your computer, google it!
• Type in the console in the left down screen help.search(”normal distribution”).
• Type help (rnorm).
• Type ?rnorm
• How is the binomial distribution?
Simulate a sample fo 200 value of a Bernoulli variable with probability of success 0.3
(W ∼ Bernoulli(p = 0.3)).
• What is the value of µW ? How do you estimate it in R?
2 ? How do you estimate it in R?
• What is the value of σW
Simulate 200 values of Z and Y of exercise 2.
• Plot the density and distribution functions of Z and Y.
• Estimate their mean, variance and covariance.
4. Conditional probabilities. We have two discrete variables X and Y . X can take
values 1, 2, 3 corresponding to three hair styles blonde, dark and bold. The variable
Y can take two values 1 and 0 corresponding to female and male. We know that these
variables probability table is:
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X/Y
Female (1)
Male (0)
Total
Blonde (1)
0.22
0.10
0.32
Dark (2)
0.29
0.18
0.47
Bold (3)
0.01
0.20
0.21
Total
0.52
0.48
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• What is the conditional probability P (X = 1|Y = 0)?
• What is the P (X = 1)?
• P (Y = 0)?
• What is the joint probability P (X = 1, Y = 0)?
• Are X and Y independent?
5. Linear models, matrix algebra and R.
• Create a script called ”myfirstR.R” with the following commands:
Y=numeric(2)
X=matrix(1, 2,2)
X[1,2]=3
X[2,2]=4
beta= c(2, 3)
Y=X%*%beta
det(X)
Y
solve(X, Y)
solve(X)%*%Y
library(fBasics)
inv(X)%*%Y
• Understand what each command does.
• Now run the following commands:
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X2= matrix(c(1, 3, 2, 6), nrow=2, byrow=T)
det(X2)
beta2=solve(X2)%*%Y
• Why cannot we find a solution to this problem?
• Click in the menu ”Session/Set Working directory/To Source File Location”.
What happens?
• Put your cursor next to the first line of your code and click on the button ”Run”
on the right side of this screen.
• Click the button ”Source” instead.
6. Create a script (File/New/R script) with the following commands:
X=matrix(1, 1000, 3)
X[,2]=rchisq(1000, df=2)
X[,3]=rnorm(1000, 1, 0.6)
epsilon=rnorm(1000, 0, 0.2)
beta=c(2, -3, 0.4)
Y=X%*%beta+ epsilon
solve(X)%*%Y
solve(X%*%X)%*%(X%*%Y)
solve(X%*%t(X))%*%(t(X)%*%Y)
solve(t(X)%*%X)%*%(t(X)%*%Y)
model.1<-lm(Y~X)
summary(model.1)
• Understand what each command does and why some of them fail.
• Interpret the results in model.1.
7. Maximum likelihood:
Given a sample of n independent identically distributed observations y1 , y2 , . . . , yn ,
normally distributed yi ∼ N (µ, σ 2 ).
• What is the joint density function of this sample?
• What is the likelihood function of this sample?
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• What is the log-likelihood function of this sample?
• What is the score vector of the log-likelihood function?
• What is the Hessian?
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