Exercise 1.3.

Look up the help file of the cumsum() function. Apply this function on the vector of Rademacher random variables you obtained in Exercise 1.2 to create a simple random walk with 1000 steps.

Plot your random walk on a suitable graph.

What is the maximal deviation of this random walk from 0?

How many times has your random walk returned to 0?

preparation: points

set.seed (10)
steps <- rbinom (n = 1000, size = 1, prob = 1/2)
rademacher_variables <- 2 * steps - 1

plot the graph

rademacher_points <- cumsum(rademacher_variables)
plot (rademacher_points,
      xlab = "steps", ylab = "displacement", main = "The plot of \n simple random walk",
      pch = 16, col = "blue")
abline (h = 0, col = "red", lty = 2, lwd = 2)
axis (2, at = 26, label = "26")

arrows (600, 25, 710, 26, col = "salmon", lwd = 2)
text(440, 25, "maximum deviation = 26", col = "salmon", cex = 0.85)

text (700, -1.8, "37 times returned to 0", cex = 0.9)

find the maximum deviation from 0

max(abs(rademacher_points))

## [1] 26

find the times returned to 0

sum(rademacher_points == 0)

## [1] 37

# or
table(rademacher_points)

## rademacher_points
## -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1 
##   6  12  15  18  18  24  25  18  11   6   5   8  15  19  17  24  36  35  29  33 
##   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19 
##  37  44  49  41  45  45  39  30  17  16  14  14  20  23  24  24  29  26  21  21 
##  20  21  22  23  24  25  26 
##  14   9   6   6   7   4   1

## using a table can also reach the result: 37 0s