Probability for Statistics / Elements of Probability — Lab Assignment 2022

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Probability for Statistics / Elements of Probability — Lab

Assignment 2022

Question 1

Your lecturer loves chocolate and has two boxes of chocolates in her o ce, one in the upper drawer and one in the lower drawer of her desk. Whenever she craves for a chocolate, she selects a drawer at random and takes a chocolate from the box in that drawer. We assume that each of the boxes originally contained 20

chocolates. Suppose your lecturer opens a drawer and discovers for the rst time that the box in that drawer is empty. We let X denote the number of chocolates left in the other box.

It can be shown that the pmf of X is

P (X = x) = 、2基-so , x = 0, 1, 2, . . . , 20.

(a)

De ne the pmf of X as a function in R, then plot this pmf over its range.

(b)

Find the probability P (X ≥ 5) using the function you de ned in (a).

(c)

Use the sample(. . . ) function to generate 10000 observations from the pmf of X. Assign the results to a variable.

(d)

Use the observations generated in (c) to obtain an estimate of P (X ≥ 5), and compare your answer with what you found in (b).

My answer:

(e)

Find the mean of X , E (X ), using the function you de ned in (a).

(f)

Use the observations generated in (c) to obtain an estimate of E (X ), and compare your answer with what you found in (e).

My answer:

(g)

Find E[(X + 1)-l ] both using the function you de ned in (a) and using the observations generated in (c), and compare your two answers.

My answer:

Question 2

For this question, we require the MosaicCalc package.

# This code installs the package only if it is not already installed, and loads it.

if( ! ( “mosaicCalc” %in% installed.packages()[,”Package”])){

install.packages(“mosaicCalc” , repos = “https://cran.ms.unimelb.edu.au/ “) }

library(“mosaicCalc”)

Let a continuous random variable X have the pdf

f(x) = ,

(a)

De ne the pdf of X as a function in R, plot that function over the support of X, and verify that -b(b) f(x)dx = 1.

(b)

Plot the cdf of X over its support.

Hint: Use the imported antiD(. . . ) function from the package MosaicCalc.

(c)

Find the mean and variance of X .

Hint: If you would like the integrate(. . . ) function to return only the numerical approximation of the integral (without the absolute error), you need to use integrate(. . . )$value.

(d)

Find P ( 2 < X < 1) using the pdf. (e) Find P ( 2 < X < 1) using the cdf. (f) Let Y = e-x . Plot the cdf of Y on its support, using the distribution function technique covered in the lectures. Question 3 Consider the continuous random variables X and Y which have the following joint pdf , 1 f(x, y) = ． 2 , 0 ≤ y ≤ x ≤ 2, (a) Find the marginal pdf fd (x) and cdf Fx (x) of X analytically. My answer: (b) Use Theorem 3.5-1 in Module 3 to simulate 10000 observations from the marginal distribution of X. Assign the results to a variable. Plot your observations in a histogram. Does the shape of the histogram mimic the shape of the pdf fd (x) My answer: (c) Repeat (a) for Y . My answer: (d) Plot the cdf of Y (e) Repeat (b) for Y . My answer: (f) Using the results of the X and Y simulations, calculate the proportion of times where, for a given observation i, X > 2Y .

(g)

Using the results of the X and Y simulations, calculate an estimate of E (X + Y).

(h)

Compute the theoretical values of P(X > 2Y) and E (X + Y), and compare them with what you obtained in (f) and (g). Do these values match with the estimations you obtained by simulations Why/why not

My answer:

(i)

BONUS QUESTION (optional: you will not loose mark if you do not answer this question, but will receive

extra mark(s) if you answwer correctly): How would you generate observations of the pair (X, Y)

My answer:

Question 4

Let X have the normal distribution N (0, 1), and let Y = ex .

(a)

Find the support of Y .

My answer:

(b)

Find the pdf of Y using the change of variable technique. You should de ne four functions throughout the process: f (x), v(y), v, (y), and the pdf of Y . Then plot the pdf of Y over (0, 5].

(c)

Find the third moment of Y , E (Y5 ), in two di erent ways: (i) using the pdf of Y , and (ii) using the known form of the mgf of a standard normal random variable.

Note: You cannot simply write xf(x) as the rst argument of the integrate(. . . ) function; instead you need to de ne a new function as xf(x).

(d)

Compute P (Y > 1). What do you conclude about the median of Y

My answer:

Question 5

Let X be a continuous random variable with the density function

, 2

f (x) = ． xl , 1 < x < 2, (a) De ne a function that computes the kth moment of X for any k ≥ 1. (b) Use the function in (a) to obtain the variance of X . (c) Find the theoretical cdf of X , and explain how you could use it to simulate a realisation/observation from X . My answer: (d) Generate a sample of 10000 observations from the distribution of X , and plot the corresponding histogram. Compare the shape of the histogram with that of the density of X (try to superimpose them on the same gure). Question 6 Let X be a Gamma distribution with mean 8 and variance 16. Let Xd , Xl , . . . , X） be n independent random variables with the same distribution as X. Let Y） = X〉/n (a) De ne a function which generates 10000 observations from Y） for any value of n. My answer: (b) Plot the histogram of the generated observations from Y） for n = 1, n = 5, n = 25. What do you observe Can you compare Y） to a known distribution when n is large Elaborate on your answer. My answer: (c) Find an appropriate normal random variable which approximates Yl4 , and plot the histogram of the generated observations from Yl4 on the same graph as the density of that normal random variable for comparison. My answer: Question 7 Let N be the number of o spring of a female Seychelles warbler (a species of birds) during a one-year period. One may assume that N has a Poisson distribution with mean 4. Seychelles warblers are known to have an adaptive sex ratio bias: on high quality territories, females produce 90% daughters. Let X be the number of daughters of one female bird during a one-year period (on a high quality territory). (a) What are the theoretical mean and variance of X My answer: (b) Conduct a two-step simulation analysis to verify these theoretical values by replacing the ‘# ’ in the code chunk with the appropriate commands (and remove the argument eval=FALSE before running the code or knitting the le). n.daughter<-c () # initialises the vector which will contain simulated values of X for (i in 1 :10000){ n.offspring<- # n.daughter<-# } # ith simulation of N # updates the vector n.daughter with the ith simulation of X mean (n.daughter) var (n.daughter) (c) What is the distribution of X Compare the results of your simulations with the true probabilities using a histogram. My answer: