Statistics Worksheet

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1. Let X have a log-normal(µ, ‡) distribution. Here is a stochastic representation of the log-normal
distribution: let Z have a standard normal distribution, and let X = e‡ Z+µ , then we say that X has
a log-normal(µ, ‡) distribution. Note that ln(X) has a normal distribution with mean µ and standard
deviation ‡.
Interpretation of the parameters:
• eµ = eE[ln(X)] is called the geometric mean of X.
Ô
• e‡ = e V [ln(X)] is called the geometric standard deviation of X.
• The mean of X is E[X] = eµ+‡
2
/2
.
Suppose that we have a random sample X1 , . . . , Xn from a log-normal population. The sample mean
qn X̄ =
qn
(1/n)
ln(Xi )
µ+‡ 2 /2
i=1
(1/n) i=1 Xi is an estimator of the mean e
, while the sample geometric mean G = e
µ
is an estimator of the geometric mean e .
Consider µ = 0, and ‡ = 1. Perform a simulation study to compare the efficiency of the estimation of the
geometric mean and of the mean? Use n = 15, 20, 25, 30, 35, 100. We will say that the estimator with the
smallest mean squared error is the most efficient estimator. Also give the estimated MSE and estimated
standard error of the MSE for both estimators.
2. Consider the continuous distribution with pdf
f (x) = (1/4)| sin(x)|, ≠ fi < x < fi. ; sin(x)/4, 0

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