Stats 426 Spring 2021
Homework 1 Due 5-14 by 10:00 pm EST
美国统计代上网课 Solve the problems in the spaces provided and save as a single PDF. Then upload the PDF to Canvas Assignments by the due date.
Solve the problems in the spaces provided and save as a single PDF. Then upload the PDF to Canvas Assignments by the due date. The recommended procedure is to download and print the homework (11 pages). Fill in your solutions. Then scan the document and upload to Canvas Assignments. If this is not feasible for you, you may solve the problems on your paper, scan your solutions, then upload to Canvas. Neatness and presentation are important. Late homework not accepted. Show all work.
Total points: 30
No need to show any work. Relate to the gamma distribution. (0.5 point)
2)Suppose the random variable X has the density givenby
a)Find E[X]. (0.5point)
b)Find V ar(X). (1point)
c)Find the cdf. Define for all real numbers. (1.5 points) 美国统计代上网课
3)Suppose X ∼ U(−1, 1). Find
a) P (|X| > ). (0.5 point)
b) the pdf of Y = X2. Make sure to define the support of the density. (1.5 points)
4)Suppose that X ∼ Exp(1). For α > 0, β > 0, and −∞ < ν < ∞,find
a)the pdf of Y = αX β + ν. Make sure to define the support of the density function. (1.5points)
b)E[Y ]. (1.5points)
5)Let X be a random variable with density 美国统计代上网课
a)Find E[X]. (1point)
b)Find V ar(X). (1point)
c)Find the cdf. Define for all real numbers. (1.5 points)
d)Find the moment generating function (mgf). Define the values t for which the mgf exists. (2points)
6)Suppose that X and Y are jointly continuous random variables suchthat X U (0, 1) and 美国统计代上网课
fY |X(y|x) = 1 x < y < x + 1.
Find P (X + Y < 1). (1.5 points)
7)SupposeX ∼ U (0, 1) and that conditional on X being observed, Y | x ∼ U (0, x).
a)Findthe joint density f (x, y). Define the support of the joint density (1 point)
b)Find the marginal density fY(y). Define the support of the density function. (1.5 points)
c)Calculate E[Y ] and V ar(Y ). (1.5points)
(Hint: Try using Theorem A page 149 and Theorem B page 151 in text)
d)Calculate Cov(X, Y ). (2points)
8)The joint density function of X and Y is givenby 美国统计代上网课
a)Findthe conditional density f (x| y). Define the support with respect to (1 point)
b)Findthe conditional density f (y| x). Define the support with respect to (1 point)
c)Find the density function of Z = XY . Define the support of the (2points)
9)Let X be a random variable having finite expectation µ and variance σ2. Let g be a twice differentiable 美国统计代上网课
Hint: Expand g in a Taylor series about µ. Use the first three terms and ignore the remainder. (1 point)
b)LetX be a Poisson random variable with mean λ. Show that if λ is not too small,
Hint: Use the result from part a) to approximate E[√X]. (1.5 points)
10)In class we considered the distribution of the random variables X andY 美国统计代上网课
a)Findthe inverse cdf of Y . That is, find FY−1(y). (1.5 points) 美国统计代上网课
We derived an algorithm for generating random variables from the joint distribution. The code to generate 1,000 observations in R is given below. Copy and paste the following code into R. Comments are preceded by the # sign.
u <- runif(1000) #generate 1,000 U(0,1) random variables y <- 1-(1-u)^(2/3) #the inverse cdf method to generate y x <- runif(1000,-sqrt(1-y),sqrt(1-y)) #x|y is U(-sqrt(1-y),sqrt(1-y)) plot(x,y) #scatterplot of x and y
b)Doesit look like the points are uniformly distributed under the function y = 1 − x2 for−1 ≤ x ≤ 1? (0.5 point) 美国统计代上网课
It is also true that x contains a sample of 1,000 random variables generated from the marginal fX(x) and y contains a sample of 1,000 random variables generated from the marginal fY (y). Just turn in the scatterplot of x and y. In a Windows environment you should be able to right click in the plot area to save it or print it.
Optional problems to try for practice (taken from your text, do not turn in)
Page 67: 40ab
Page 69: 54, 55, 56, 58, 59, 61, 66, 67abc
Page 108: 14
Page 112: 43
Page 167: 13
Page 168: 14
Page 170: 44, 50
Page 171: 55, 56
More optional problems to try for practice (do not turn in)
1)Let X be a random variable with cumulative distribution function(cdf) 美国统计代上网课
a)Find P (X >0.5).
b)Find the median. This is the value msuch tha
c)Find the standard deviation.
d)Let Y = √X. Find the pdf and the cdf of Y.
2)LetX1, X2, . . . Xn be iid from Exponential(1/θ). That is
Calculate E[nX(1)] where X(1) is the smallest order statistic.
3)LetX1, . . . , Xn be independent and identically distributed (iid) random variables each with marginal density 美国统计代上网课
Find the pdf of the random variable
4)LetZ1 and Z2 be independent N (0, 1) random Calculate Cov(Z1 + Z2, Z1 − Z2).
5)IfX ∼ N (0, σ2), find the pdf of Y = |X|. Make sure to define the support of the density function (the values of y where fY (y) > 0).
6)SupposeX N (µ, σ2). Find the pdf of Y = eX. This is known as the lognormal distribution. Make sure to define the support of the density function (the values of
y where Y (y) > 0). 美国统计代上网课
7)Jill’sbowling scores are approximately normally distributed with mean 170 and stan- dard deviation 20, while Jack’s scores are approximately normally distributed with mean160 and standard deviation If Jack and Jill each bowl one game, then assuming that their scores are independent random variables, approximate the probability that
a)Jack’s score is higher.
b)the total of their scores is above 350.
8)The joint density of X and Y is givenby
f (x, y) = x e−(x+y) x > 0 y > 0
a)Determine if X and Y are independent.
If instead, f (x, y) were given by
f (x, y) = 2 0 < x < y 0 < y < 1
b)Determine if X and Y are independent. 美国统计代上网课
c)Calculate Cov(X, Y).
9)If X1 and X2 are independent U (0, 1) random variables, then the joint pdf of the order statistics (X(1), X(2)) is
f(X(1), X(2))(x1, x2) = 2 0 < x1 < x2 < 1
For n ≥ 1 find P (nX(1) > X(2)). Calculate
10)Let X1, X2be iid N (0, 1). Let
Y1 = g1(X1, X2) = X1 + X2 Y2 = g2(X1, X2) = X1 − X2
a)Find thejoint pdf f (y1, y2).
b)Are Y1and Y2 independent?
c)What are the marginal distributions of Y1and Y2?
11)Supposethat the conditional density of X given Y = y has an exponential distribution with parameter y. 美国统计代上网课
In addition, let Y Gamma(α, λ). Find the marginal density f (x) (this density will depend on parameters α and λ).
12)Suppose X has the density givenby
f (x) = ex−ex − ∞ < x < ∞
a)Find the cdfFX(x).
The density of a distribution truncated on the left by a and the right by b is given by
b)Find the truncated density when a = 0 and b =∞.
13)Suppose the random variable X has the density givenby
fX(x) = λ x−(λ+1) x > 1 , λ > 0 Find the density of Y = log X. Do you recognize this density? 美国统计代上网课
14)The distribution givenby
is called the Rayleigh distribution.
a)FindE[X]. b) Find V ar(X).