Question

X₁ is a binomial random variable with n = 100 and p = 0.8, while X₂ is a binomial random variable with n = 100 and p = 0.2. The variables are independent. The profit measure is given by P = X₁ / X₂. Run N = 10 simulations of X₁ and X₂ to simulate the distribution of P, and then find both the variance and the semi-variance.

Answer 1

Given X1 and X2 are binomial random variable with n1 = 100, p1 = 0.8 and n2 = 100, p2 = 0.2 respectively.

Using R to simulate the distribution of P = X1/X2.

Creating variables in R as follows:

n1 = 100; p1 = 0.8
n2 = 100; p2 = 0.2

Simulating 10 values of both X1 and X2 using rbinom() function:

x1 <- rbinom(10, 100, 0.8) ; x1
[1] 78 84 78 81 80 74 80 84 73 79

x2 <- rbinom(10, 100, 0.2) ; x2
[1] 16 13 19 15 17 19 20 17 27 16

Calculating values of P:

P = x1/x2 ; P
[1] 4.875000 6.461538 4.105263 5.400000 4.705882 3.894737 4.000000
[8] 4.941176 2.703704 4.937500

Calculating variance of P using var() function:

var(P)
[1] 1.01339

Now to calculate semi-variance we use the formula:

Where:

n = the total number of observations below the mean

r_t = the observed value (observations below mean)

Average = the mean of the dataset

First we calculate the mean using mean() function in R:

mean(P)
[1] 4.60248

Then we calculate r_t as follows:
rt = P[P < mean(P)] ; rt
[1] 4.105263 3.894737 4.000000 2.703704

Calculating n (the total number of observations below the mean):

n = length(rt) ; n
[1] 4

Finally calculating semivariance:

semi_var = sum((mean(P) - rt)^2)/n
semi_var
[1] 1.179115

(The R code is in bold)