Project 1
Sample mean
• μ is the population mean and θ=μ. If θ = X
⌢ ⌢
and X is normally distributed with mean 0
and std deviation 1 ( X~ N(0,1) ), what is the distribution of θ
⌢
look like? The number of
trials is n=6.
• Run a Monte Carlo simulation with 2,000 replications.
• Check whether the theory is right. How?
Sample standard deviation (S)
• Situation: Standard normal data (~N(0,1)), for n=10 trials,
• Use Monte Carlo simulation with 1,000 reps to answer these questions
– What is the distribution of S in this situation?
– What is the bias of S in this situation?
– What is the standard error (SE) of S in this situation?
– What is the SE of S when n=4 instead of 10?
Central limit theorem
• Generate 100 samples of exponential distribution with mean rate 2 (shown by EXPO(2))
for n=4 trials, take their averages and plot the histogram, Do X bars look Normal?
• Generate 100 samples of EXPO(2) for n=20 trials, take their averages and plot the
histogram, Do X bars look Normal?
• Genarate 100 samples of Uniform (0, 1) for n=4 trials, take their averages and plot the
histogram, Do X bars look Normal?

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