lista de exercicios estatistica, Exercícios de Estatística

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• Homework stat
  • spring
    • 1-
  • Version–B-
• 2016-05-09-21-15-
• Homework 1 - January
• Homework 2 - February
• Homework 3 - February
• Homework 4 - February
• Homework 5 - February
• Homework 6 - March
• Homework 7 - March
• Homework 8 - April
• Homework 9 - April
• Homework 10 - April
• Homework 11 - April
• Homework 12 - May
• Homework 13 - May
• Homework 14 - May

Homework 1 - January 25

Problem 1.

[CB, Exercise 3.17, page 130] Establish a formula similar to (3.3.18) for the gamma distribution. If X ∼ gamma(α, β), then for any positive constant ν ,

E Xν^ =

βν^ Γ(ν + α) Γ(α)

Problem 1. [CB, Exercise 3.20, page 131] Let the random variable X have pdf^1

f (x) =

2 π

exp

− x^2 / 2

(^1) X = |Z|.

The Pareto(α, β) distribution has pdf

f (x) =

βαβ x1+β^

1(α < x < ∞), α, β > 0.

i) Verify that f is a pdf. ii) Derive the mean and the variance of this distribution. iii) Prove that the variance does exist if β ≤ 2.

Problem 1. [CB, Exercise 3.24abcd, page 131] Many named distributions are special cases of the more common distributions already discussed. For each of the following named distributions derive the form of the pdf, ver- ify that it is a pdf, and calculate the mean and variance.

i) If X is Exp(β) then X

1 γ (^) ∼ Weibull(γ, β) where γ > 0.

ii) If X is Exp(β) then

2 X

β

has the Raylight distribution.

iii) If X ∼ gamma(a, b), then Y =

X

has the inverted gammaIG(a, b) distribu- tion^4.

iv) If X is gamma

, β

then

X

β

has the Maxwell distribution.

Problem 1. [CB, Exercise 3.25, page 131] Suppose the random variable T is the length of the life of an object. The hazard func- tion hT associated with random variable T is

(^4) This distribution is useful in Bayesian estimation of variance, see Exercise 7.23.

ii) Show that, under the stated conditions,

(2) FT (t) = 1 − exp

∫ (^) t

0

h(s)ds

iii) Explain that (2) defines a proper dis- tribution whenever h ≥ 0, h is measur- able and

0 h(s)ds^ =^ ∞.

Problem 1. [CB, Exercise 3.26, page 132] Verify that the following pdfs have the indi- cated hazard functions

i) If T is Exp(β) , then hT (t) =

β

ii) If( T is Weibull(γ, β) , then hT (t) = γ β

tγ−^1.

iii) If T is logistic(μ, β) , that is

FT (t) =

1 + exp

t − μ β

then hT (t) = (1/β)FT (t).

Homework 2 - February 1

Problem 2.

[CB, Exercise 3.28-c, page 132] Show that each of the following families is an exponential family.

i) Normal family with either parameter μ or σ^2 known. ii) Gamma family with either parameter α or β or both known. iv) Poisson family. v) Negative binomial family where p ∈ (0, 1).

Problem 2. [CB, Exercise 3.29, page 132] For each family in Exercise 3.28, describe the natural parameter space.

Problem 2. [CB, Exercise 3.31, page 132] In this exercise we will prove Theorem 3.4. (page 112).

i) Start from the equality ∫ f (x|θ)dx =

h(x)c(θ)exp

( (^) ∑k

i=

wi(θ)t

ii) Differentiate the above equality a sec- ond time; then rearrange to establish (3.4.5)^7

Problem 2. [CB, Exercise 3.32, page 133]

(^7) The fact that d (^2) log g(x) d^2 x = −

( (^) d log g(x) dx

) 2

1 g g′′(x) may be helpful.

Problem 2. [CB, Exercise 3.33ac, page 133] For each of the following families:

i) Verify that it is an exponential family: iii) Sketch a graph of the curved parameter space.

a) N (θ, θ). b) N (θ, aθ^2 ), a is known. c) gamma(α, α−^1 ).

Problem 2. [CB, Exercise 3.38, page 133]

Let Z be a random variable with pdf f. Define zα 8 to be a number satisfying

α = P (Z > zα) =

zα

f (z)dz.

Show that if X is random variable with pdf

σ−^1 f

σ−^1 (x − μ)

and if xα^ def = σzα + μ, then P (X > xα) = α^9.

(^8) This is the α-quantile. (^9) Thus if a table of zα values were available, then

values of zα could be easily computed for any mem- ber of the location-scale family.

Homework 3 - February 8

Problem 3.

[CB, Exercise 4.1, page 192] A random point is distributed uniformly on the square with vertices (1, 1), (1−1), (− 1 , 1), and (− 1 ,. − 1). That is the joint pdf is

f (x, y) =

on the square. 12 Determine

the probabilities of the following events,

i) X^2 + Y 2 < 1. ii) 2 X − Y > 0. iii) |X + Y | < 2.

(^12) f (x, y) = 4− 11 ((x, y) ∈ square), the last term is an indicator.

Problem 3. [CB, Exercise 4.4, page 192] A pdf is defined by 13

f (x, y) = C(x+2y) 1(0 < y < 1)1(0 < x < 2).

i) Find the value of C. ii) Find the marginal distribution of X. iii) Find the joint cdf of X and Y. iv) Find the pdf of the random variable Z = 9(X + 1)−^2.

(^13) The two terms on the right hand side are indi- cators, e.g. 1(0 < y < 1) = 1(0,1)(y).