数理统计代考 STATS 426代写

STATS 426

Winter, 2021

数理统计代考 Attempt all questions. Give short and speciftc answers. Show intermediate steps if there are any – to gain partial credits.

Exam #1 (Evening Exam-Schedule 2; due at 8pm)

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1) Show your work and derivations, except for the questions in Problem #1 (Multiple Choices). A correct answer without derivations/procedures in other problems would get you NO Forthe sub-problems in Problem #1, you don’t need to give any explanation if you are sure your answer is correct. If you are not sure, a correct and relevant explanation could give you partial credits.

2)Make sure you upload your answer file to Canvas site before the given deadline; every additional minutes will cost you additional points until the score reaches 0.

3) Attempt all questions. Give short and speciftc answers. Show intermediate steps if there are any – to gain partial credits. 数理统计代考

4) Give only one answer to each question – if you give multiple potential answers, the worst answer will begraded.

• 5) During the exam time, you are not allowed to have interactions about the exam or course related materials with anyone other than the instructor. It would be considered cheating if you do so. Please complete the exam by yourself.

6) There are totally 10 questions in this exam. You have 60 minutes to complete the exam and a20-minute window to upload the pdf file.

GOOD LUCK

PROBLEM # 1:Multiple Choice — Choose the correct answer. Each multiple choice question is worth (4) points. 数理统计代考

1.1_____ If X₁ and X₂ both follow a Gamma distribution with parameter α > 0 and λ > 0 then

(A) P(X₁ < 0) changes with the value of α (B) S = X₁ + X₂ is also a Gamma random variable

1.2________The number of offspring of an organism is a discrete random variable with mean α and variance δ, where α > 0 and δ > Each of its offspring reproduces in the same manner. Suppose we have N₁ = 5 organism at the first generation and N₂, and N₃ numbers of offspring at the second and third generation, respectively. Which of the following is correct.

(A) E(N₂) = 5α. (B) V ar(N₂) = 5²δ.

1.3________Let X and Y be 2 random variables: E(X) = 0, var(X) = σ²> 0 and Y given X follows N (X, α), where α is a constant.

Let var(Y ) = σ² . Then (A) α > 0 (B) σ² = σ²+ α (C) X and Y are Y X Y positively correlated (D) None of the above is always true.

1.4________Let X be a continuous random variable with pdf, f , and a < b are real numbers. Is the 000following statement correct? “P (a ≤ X ≤ b) = f (a) + ∫ b f (s)ds + f (b).” 数理统计代考

(A) Yes (B) No

1.5________Let X, Y be joinly normally distributed with parameters µ_X , µ_Y , σ² , σ² , and ρ. Let X Y fX|Y (x|y₀) denote the conditional pdf of X given Y = y₀. Let E(X|Y = y₀) be the mean for fX|Y (x|y₀).Also, σ²> 0 and σ² > 0, then E(X|Y = y₀)

(A) ≥ µ_X if y₀ > µ_Y and ρ > 0 (B) ≤ µ_X if y₀ < µ_Y and ρ < 0

PROBLEM # 2: 数理统计代考

Consider tossing a coin, with outcomes, H & T, and P(H) = π₀. Each time, if you get a head, you pick up 3 blue balls and if you get a tail, you pick up 1 red ball.

2.1(5 pt) Suppose you toss a coin independently 2 times and let Y = the toal number of balls (no matter it is red or blue). Please provide the pmf function for Y and show it is a pmf function.

2.2(5 pt) Suppose you toss a coin independently n times and let Y = the toal number of balls. What isthe distribution of Y when n → ∞? Please show how you obtain your answer and report the distribution and its parameters.

PROBLEM # 3: 数理统计代考

Let X₁ and X₂ be two independent random variables with pdf, f_X (x), and f_X (x) =θ²e^−θ2x, when x ≥ 0; f (x) = 0, otherwise. Also let V = X − X .

3.1(5 pt) What would be the pdf of (V )? Justify that your answer is a pdf.

3.2(5pt) Suppose we generate 3 random numbers from Uniform distribution, U[0,1], and they are 0.1, 0.24and 0.36. Using these 3 numbers, how should we obtain 3 random numbers generated from the distribution of V ?

PROBLEM # 4:

Suppose Y_n ∼ Poisson(θ_n) where θ_n → ∞, witn n → ∞. Please use the moment generating functions of Y_n to show that when n → ∞,

That is Y_n converge in distribution to a normal distribution. The Poisson mgf is exp{θ(exp(t) − 1)}.

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