Use the Monte Carlo method to estimate the mean, variance, and 2.5th and 97.5th percentiles of a gamma distribution G(2, 4). Compare the estimates with the true values and comment how the sample size affects the estimates.

Given information: shape = 2, scale = 4 for the gamma distribution under study (rate = 1/scale = 1/4)

Let us take different number of samples to study effect of sample size on estimates

R-code and Solution :

ss = c(50,100,200,500,1000,10000,100000)

sh = 2

sc = 4

i = 0

means = list()

vars = list()

quant_025 = list()

quant_975 = list()

for(s in ss)

{

i = i+1

sample_series = rgamma(n = s, shape = sh, scale = sc)

means[i] = mean(sample_series)

vars[i] = var(sample_series)

quant_025[i] = qgamma(p = c(0.025), shape = sh, scale = sc)

quant_975[i] = qgamma(p = c(0.975), shape = sh, scale = sc)

}

i = 0

true_mean = sh*sc

# 8

true_variance = sh*sc^2

# 32

# Means:

## True mean = 8

## as.matrix(means)

# [1] 9.82338

# [2] 8.970444

# [3] 7.859794

# [4] 8.204468

# [5] 7.613365

# [6] 8.055619

# [7] 7.990096 --> As sample size increases, the estimated mean tends to the theoretical mean

# Variances:

## True variance = 32

## as.matrix(vars)

# [1] 35.64035

# [2] 54.33689

# [3] 29.01417

# [4] 32.25097

# [5] 27.00697

# [6] 32.92603

# [7] 32.10989 --> As sample size increases, the estimated variance tends to the theoretical variance

# Quantiles:

## as.matrix(quant_025) and as.matrix(quant_975)

## 2.5% 97.5%

# [1] 0.9688371 22.28657

# [2] 0.9688371 22.28657

# [3] 0.9688371 22.28657

# [4] 0.9688371 22.28657

# [5] 0.9688371 22.28657

# [6] 0.9688371 22.28657

# [7] 0.9688371 22.28657

R-code and Solution

Step-by-step explanation:

step-by-step explanation:

if you want to solve percent error follow these steps.

1. subtract the accepted value from the experimental value.

2. take the absolute value of step 1.

3. divide that answer by the accepted value.

4. multiply that answer by 100 and add the % symbol to express the answer as a percentage.

if you want to solve percent of change just do this.

1. work out the difference (increase) between the two numbers you are comparing.

2. divide the increase by the original number and multiply the answer by 100. if your answer is a negative number then this is a percentage decrease.

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