Demo Week 0

Question 1

Generate 100 random vectors \(X_1,\ldots, X_{n}\) (\(n=100\)) from the bivariate normal distribution \(\mathcal{N}(\mu,\Sigma)\), where \(\mu=(1,2)^\intercal\) and \[ \Sigma = \begin{bmatrix} 1 & 0.5 \\ 0.5 & 2 \end{bmatrix}. \] Compute the sample mean vector and the sample covariance matrix: \[ \hat \mu = \frac{1}{n} \sum_{i=1}^n X_i ~~\mbox{ and }~~ \hat \Sigma = \frac{1}{n} \sum_{i=1}^n (X_i- \hat \mu) (X_i - \hat \mu)^\intercal . \]

Answer:

This can be done using the function in the package:

library(MASS)
Sigma <- matrix(c(1,0.5,0.5,2),2,2)
mu <- c(1, 2)
X <- mvrnorm(n = 100, mu, Sigma) 
# n by 2 data matrix X: each row represents a sample
head(X)

##            [,1]        [,2]
## [1,]  1.5095904  1.19634149
## [2,]  1.5232664  3.29298655
## [3,]  0.8176725  0.47466701
## [4,]  1.4941909  1.38292277
## [5,] -0.3138479 -0.02347093
## [6,]  2.0448483  3.04146990

# Sample mean vector
smean <- colMeans(X)

# Sample covariance matrix
n <- dim(X)[1]
X0 <- X - matrix(smean, nr = n, ncol = 2 , byrow = TRUE) 
# Centered data matrix X0: each row represents a sample vector minus the sample mean
scov <- t(X0) %*% X0 /n

Question 2

Write an function that outputs the first \(n\) terms in the Fibonacci sequence, where \(n\) is the only input value of the function.

Answer:

# construct the function
fibonacci <- function(n)
{
x = numeric(n)
x[1:2] = c(1,1)
for(i in 3:n) x[i] = x[i-1] + x[i-2]
return(x)
}
# test the function
fibonacci(10)

##  [1]  1  1  2  3  5  8 13 21 34 55

Instead, using echo = FALSE in the code chunk prevents printing of the R code that generated the function