# 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