Consider a distribution with parameter λ >0 that has density f(x)= x^4/(24 λ^5) e^(-x/λ). You test the hypothesesH0:t...
Consider a distribution with parameter λ >0 that has density f(x)= x^4/(24 λ^5) e^(-x/λ).
You test the hypothesesH0:test λ =1 vs λ ≠ 1 by using the test
Test: I Xbar-5 I > c. Find the smallest threshold C for an asymptotic level α
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Consider a distribution with parameter λ >0 that has density f(x)= x^4/(24 λ^5) e^(-x/λ). You test the hypothesesH0:t...
5. The Exponential(A) distribution has density f(x) = for x<0' where λ > 0 (a) Show/of(x)...
5. The Exponential(A) distribution has density f(x) = for x<0' where λ > 0 (a) Show/of(x) dr-1. (b) Find F(x). Of course there is a separate answer for x 2 0 and x <0 (c Let X have an exponential density with parameter λ > 0 Prove the 'Inemoryless" property: P(X > t + s|X > s) = P(X > t) for t > 0 and s > 0. For example, the probability that the conversation lasts at least t...
The Poisson distribution with parameter λ has the mass function defined by p(x) = λ x...
The Poisson distribution with parameter λ has the mass function defined by p(x) = λ x e −λ/x! if x is a nonnegative integer (and 0 otherwise). Find the probability it assigns to each of the following sets: a. [0, 2) b. (−∞,1] c. (−∞,1.5] d. (−∞, 2) e. (−∞,2] f. (0.5, ∞) g. {0, 1, 2} Find the CDF of the uniform distribution on (0,1).
8. The Gamma(a, A) distribution has density f(x)(a) where for a0' a 0 and A >...
8. The Gamma(a, A) distribution has density f(x)(a) where for a0' a 0 and A > 0 (a) Showfx,of(x) dr-1. Recall !"(a)-C"rtta-idt. (b) If X has a gamma distribution with parameters α and λ, find a general expression for E(Xk). (Answer: ) (c) Use your answer to the last question to find Var(X). The identity「(α + 1) a「(a) will help.
You test the hypotheses Ho A 1 vs H1 : > # 1 by using the test =1 (|X, 5> Can) Find the smallest threshold Can so...
You test the hypotheses Ho A 1 vs H1 : > # 1 by using the test =1 (|X, 5> Can) Find the smallest threshold Can so that the test has asymptotic level a. choosing a test so that Can does not depend on the estimate of A Note that here we are (If applicable, enter Phi(z) for the cdf (z) of a normal variable Z, q(alpha) for the quantile qa. Recall the convention in this course that P (Z...
Problem 1-5 1. If X has distribution function F, what is the distribution function of e*?...
Problem 1-5
1. If X has distribution function F, what is the distribution function of e*? 2. What is the density function of eX in terms of the densitv function of X? 3. For a nonnegative integer-valued random variable X show that 4. A heads or two consecutive tails occur. Find the expected number of flips. coin comes up heads with probability p. It is flipped until two consecutive 5. Suppose that PX- a p, P X b 1-p, a...
Let X1,..., X, be an i.i.d. sample from a Rayleigh distribution with parameter e > 0:...
Let X1,..., X, be an i.i.d. sample from a Rayleigh distribution with parameter e > 0: f(x\C) = e ==/(20?), x20 (This is an alternative parametrization of that of Example A in Section 3.6.2.) a. Find the method of moments estimate of e. b. Find the mle of C. Find the asymptotic variance of the mle.
The number of medical emergency calls per hour has a Poisson distribution with parameter λ. Calls...
The number of medical emergency calls per hour has a Poisson distribution with parameter λ. Calls received at different hours are considered to be independent. Emergency calls X1 ,…, Xn for n consecutive hours has the same parameter λ. a) What is the distribution of Sn = ∑ Xi ? b) Provide Normal approximation for the distribution of Sn . c) Provide maximum likelihood estimation of λ. Calculate variance and bias of MLE. d) Calculate Fisher information and efficiency of...
Recall that a discrete random variable X has Poisson distribution with parameter λ if the probability...
Recall that a discrete random variable X has Poisson
distribution with parameter λ if the probability mass function of
X
Recall that a discrete random variable X has Poisson distribution with parameter λ if the probability mass function of X is r E 0,1,2,...) This distribution is often used to model the number of events which will occur in a given time span, given that λ such events occur on average a) Prove by direct computation that the mean of...
Expected Frequency is found by using the Poisson distribution -λ χ)-__ where λ-[0(24) + 1(30) +...
Expected Frequency is found by using the Poisson distribution -λ χ)-__ where λ-[0(24) + 1(30) + 2(31) + 3(11) + 4(4))/100-1.41 2. ChieX Table Valuc Observed Frequency 24 30 4 1) The variable of interest is the form of the distribution for X 2) Ho: The form of the distribution is Poisson 3) H: The form of the distribution is not Ppisson f (x-3) = ? 5) The test statistic is
In question 5, f(x) = λ*exp(-λx), for x greater or equal to 0, and zero otherwise....
In question 5, f(x) = λ*exp(-λx), for x greater or
equal to 0, and zero otherwise.
9. Let X have an exponential distribution with λ = 1 (see Question 5), and let Y = log(X). Find the probability density function of Y. Where is the density non-zero? Note that in this course, log refers to the log base e, or natural log, often symbolized In. The distribution of Y is called the (standard) Gumbel, or extreme value distribution. 2
5. The Exponential(A) distribution has density f(x) = for x<0' where λ > 0 (a) Show/of(x) dr-1. (b) Find F(x). Of course there is a separate answer for x 2 0 and x <0 (c Let X have an exponential density with parameter λ > 0 Prove the 'Inemoryless" property: P(X > t + s|X > s) = P(X > t) for t > 0 and s > 0. For example, the probability that the conversation lasts at least t...
8. The Gamma(a, A) distribution has density f(x)(a) where for a0' a 0 and A > 0 (a) Showfx,of(x) dr-1. Recall !"(a)-C"rtta-idt. (b) If X has a gamma distribution with parameters α and λ, find a general expression for E(Xk). (Answer: ) (c) Use your answer to the last question to find Var(X). The identity「(α + 1) a「(a) will help.
You test the hypotheses Ho A 1 vs H1 : > # 1 by using the test =1 (|X, 5> Can) Find the smallest threshold Can so that the test has asymptotic level a. choosing a test so that Can does not depend on the estimate of A Note that here we are (If applicable, enter Phi(z) for the cdf (z) of a normal variable Z, q(alpha) for the quantile qa. Recall the convention in this course that P (Z...
Problem 1-5
1. If X has distribution function F, what is the distribution function of e*? 2. What is the density function of eX in terms of the densitv function of X? 3. For a nonnegative integer-valued random variable X show that 4. A heads or two consecutive tails occur. Find the expected number of flips. coin comes up heads with probability p. It is flipped until two consecutive 5. Suppose that PX- a p, P X b 1-p, a...
Let X1,..., X, be an i.i.d. sample from a Rayleigh distribution with parameter e > 0: f(x\C) = e ==/(20?), x20 (This is an alternative parametrization of that of Example A in Section 3.6.2.) a. Find the method of moments estimate of e. b. Find the mle of C. Find the asymptotic variance of the mle.
Recall that a discrete random variable X has Poisson
distribution with parameter λ if the probability mass function of
X
Recall that a discrete random variable X has Poisson distribution with parameter λ if the probability mass function of X is r E 0,1,2,...) This distribution is often used to model the number of events which will occur in a given time span, given that λ such events occur on average a) Prove by direct computation that the mean of...
Expected Frequency is found by using the Poisson distribution -λ χ)-__ where λ-[0(24) + 1(30) + 2(31) + 3(11) + 4(4))/100-1.41 2. ChieX Table Valuc Observed Frequency 24 30 4 1) The variable of interest is the form of the distribution for X 2) Ho: The form of the distribution is Poisson 3) H: The form of the distribution is not Ppisson f (x-3) = ? 5) The test statistic is
In question 5, f(x) = λ*exp(-λx), for x greater or
equal to 0, and zero otherwise.
9. Let X have an exponential distribution with λ = 1 (see Question 5), and let Y = log(X). Find the probability density function of Y. Where is the density non-zero? Note that in this course, log refers to the log base e, or natural log, often symbolized In. The distribution of Y is called the (standard) Gumbel, or extreme value distribution. 2
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