Question

Question 6. Using only a random number generator that simulates a random number between [0,1], simulate 1,000 iterations of the following random variables:

a) X ~ exp(5)

b) X such that f(x) = 3x2 , 0 ≤ x ≤ 1

For both (a) and (b), plot a histogram of your simulated random variable. Estimate ?̅ and s2. Compare these values to E(X) and Var(X).

Answer 1

GIVEN THAT

Using only a random number generator that simulates a random number between [0,1].

(a) exponential (5)

Inverse tranformation

U=

-5x=log(l-U)

x=

(b) we have

=

=3

The inverse transformation

U=

x=

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R Graphics. Device 2 (ACTIVE) histogram of mean1 250 200 200 150 150 Frequency 100 BE 100 50 0 0.18 0.19 0.20 0.21 0.22 mean1 CS Scanned with CamScanner

R Graphics: Device 2 (ACTIVE) histogram of variance1 200 150 Frequency 100 0.030 0.035 0.040 0.045 0.050 variance 1 Scanned with CamScanner

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QR Graphics: Device 2 (ACTIVE) DX histogram of variance2 250 200 150 Frequency i 100 50 0.032 0.034 0.036 0.038 0.040 0.042 0.044 variance2 ES Scanned with CamScannen

CORR Graphics Device 2 (ACTIVE) histogram of mean2 300 250 200 Frequency 150 100 50 0.73 0.75 MM 0.76 0.77 cs Scanned with mean2 CamScanner