Question

4. consider a random process Xt Sin(TOOt+ φ). / nen/ent a 2π (a) Consider φ ~ Uniform(-π, π). Plot the process for t E (0, 1000) (b) Consider φ ~ Uniform(0,π), plot the process for t E (0, 1000)
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Answer #1

The R program and figures are added below. For t, intervals of length 0.1 are considered.

R Program

#a
time=seq(0,1000,.1)
n=length(time)
x=numeric(n)
for(i in 1:n)
{
u=runif(1,-pi,pi)
x[i]=sin(2*pi*time[i]/100+u)
}
plot(time,x,xlab="t",ylab="X_t",main="(a)",col="red")

#b
time=seq(0,1000,.1)
n=length(time)
x=numeric(n)
for(i in 1:n)
{
u=runif(1,0,pi)
x[i]=sin(2*pi*time[i]/100+u)
}
plot(time,x,xlab="t",ylab="X_t",main="(b)",col="green")

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