# Starter Problems¶

## Strang Matrix Problem¶

In [1]:
N = 10
A = zeros(N,N)
for i in 1:N, j in 1:N
abs(i-j)<=1 && (A[i,j]+=1)
i==j && (A[i,j]-=3)
end
A

Out[1]:
10Ã—10 Array{Float64,2}:
-2.0   1.0   0.0   0.0   0.0   0.0   0.0   0.0   0.0   0.0
1.0  -2.0   1.0   0.0   0.0   0.0   0.0   0.0   0.0   0.0
0.0   1.0  -2.0   1.0   0.0   0.0   0.0   0.0   0.0   0.0
0.0   0.0   1.0  -2.0   1.0   0.0   0.0   0.0   0.0   0.0
0.0   0.0   0.0   1.0  -2.0   1.0   0.0   0.0   0.0   0.0
0.0   0.0   0.0   0.0   1.0  -2.0   1.0   0.0   0.0   0.0
0.0   0.0   0.0   0.0   0.0   1.0  -2.0   1.0   0.0   0.0
0.0   0.0   0.0   0.0   0.0   0.0   1.0  -2.0   1.0   0.0
0.0   0.0   0.0   0.0   0.0   0.0   0.0   1.0  -2.0   1.0
0.0   0.0   0.0   0.0   0.0   0.0   0.0   0.0   1.0  -2.0

## Factorial Problem¶

In [2]:
function my_factorial(n)
k = one(n)
for i in 1:n
k *= i
end
k
end

my_factorial(4)
my_factorial(30)
my_factorial(big(30))

Out[2]:
265252859812191058636308480000000

## Binomial Problem¶

In [3]:
function binomial_rv(n, p)
count = zero(n)
U = rand(n)
for i in 1:n
U[i] < p && (count += 1)
end
count
end

bs = [binomial_rv(10, 0.5) for j in 1:10]

Out[3]:
10-element Array{Int64,1}:
5
5
1
5
5
2
4
5
7
6

## Monte Carlo $\pi$ Problem¶

In [7]:
n = 10000000

count = 0
for i in 1:n
global count
u, v = 2rand(2) .- 1
d = sqrt(u^2 + v^2)  # Distance from middle of square
d < 1 && (count += 1)
end

area_estimate = count / n

print(area_estimate * 4)  # dividing by radius**2

3.1420112

# Integration Problems¶

## Timeseries Generation Problem¶

In [8]:
using Plots; gr()

alphas = [0.0, 0.5, 0.98]
T = 200

series = []
labels = []

for alpha in alphas
x = zeros(T + 1)
x[1] = 0.0
for t in 1:T
x[t+1] = alpha * x[t] + randn()
end
push!(series, x)
push!(labels, "alpha = \$alpha")
end

plot(series, label=reshape(labels,1,length(labels)),lw=3)