1. Suppose that the number of red blood corpuscles in humans, denoted by X, follows a Poisson distribution whose parameter depends on the observed individual. For a person selected at random we may consider the parameter value Y as an that Y y, we have X Exp(a) random variable such that, given Poisson(y); namely, Exp(a) X | Y yPoisson(y), with Y Here a is a positive constant and the density function of Exp(a) is f(y)aeay, y > 0 (a) Find the conditional expectation E(X|Y = y), where y 0 (b) Compute the expectation E(X). Remark: You may use the law of total expectation. (c) Find the distribution of X. Remark: You may use the law of total probability: P(X k) = So P(X k|Y y)fy (y)dy

1. Suppose that the number of red blood corpuscles in humans, denoted by X, follows a Poisson distribution whose parameter depends on the observed individual. For a person selected at random we may consider the parameter value Y as an that Y y, we have X Exp(a) random variable such that, given Poisson(y); namely, Exp(a) X | Y yPoisson(y), with Y Here a is a positive constant and the density function of Exp(a) is f(y)aeay, y > 0 (a) Find the conditional expectation E(X|Y = y), where y 0 (b) Compute the expectation E(X). Remark: You may use the law of total expectation. (c) Find the distribution of X. Remark: You may use the law of total probability: P(X k) = So P(X k|Y y)fy (y)dy

Name: I need an easy to understand solution with a thorough explanation. I h
Uploaded: 2024-03-20T07:06:41.000Z
Channel: Bartleby
Description: I need an easy to understand solution with a thorough explanation. I have no good background in probability. I hope you will provide me with a very good solution, Thank you very much 1. Suppose that the number of red blood corpuscles in humans, denot

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

See similar textbooks

Similar questions

If X is a continuous random variable with X ∼ Uniform([0, 2]), what is E[X^3]?
If X1, X2, ... , Xn constitute a random sample of size nfrom a geometric population, show that Y = X1 + X2 +···+ Xn is a sufficient estimator of the parameter θ.
Suppose X, Y, Z are iid observations from a Poisson distribution with parameter λ, which is unknown. Consider the 3 estimators T1 = X + Y − Z, T2 = 2X + Y + Z 4 , T3 = 3X + Y + Z 5 . (a) Which among the above estimators are unbiased? (b) Among the class of unbiased estimators, which has the minimum variance?
Suppose that three random variables X1, X2, X3 form a random sample from the uniform distribution on interval [0, 1]. Determine the value of E[(X1-2X2+X3)2]
If the random variable X follows the uniform distribution U= (0,1) What is the distribution of the random variable Y= -2lnX. Show its limits.
X is an poisson random variable with parameter λ = 4 Calculate P {X ≤ 3}
Let X1, X2, ..., X10 be a random sample from a gamma distribution with α = 3 and β = 1/θ. Suppose we believe that theta follows a gamma-distribution with α = 28 and β = 17 and suppose we have a trial (X1 , ... , Xn ) with an observed x̄ = 28.2 What is the posterior distribution of θ What is the Bayes point estimate of θ associated with the square-error loss function?
Show that, if X1 and X2 are independent random variables with geometric distributions of aparameter p, then Y = X1 + X2 has a negative binomial distribution with parameters p andk = 2.
4.4. An individual picked at random from a population has a propensity to have accidents that is modelled by a random variable Y having the gamma distribution with shape parameter α and rate parameter β. Given Y = y, the number of accidents that the individual suffers in years 1, 2, . . . , n are independent random variables X1, X2, . . . Xn each having the Poisson distribution with parameter y. (a) Write down a function f so that the joint distribution of Y, X1, . . . , Xn can be described via P(a ≤ Y ≤ b, X1 = k1, X2 = k2 . . . Xn = kn) = Z b a f(y, k1, k2, . . . kn)dy and derive from this expression that, for your choice of f, Y has the Gamma distribution, and that conditionally on Y = y, X1, X2, . . . Xn are independent, each having the Poisson distribution with parameter y. (b) Find the conditional distribution of Y given that X1 = k1, X2 = k2, . . . , kn. (c) An insurance company has observed the number of accidents that an individual has suffered on each of n years and wishes to…
Let X1, X2, ... , Xn be a random sample, normally distributed with mean μ and variance σ2If σ2 is unknown, find a minimum value for n to guarantee, with probability 0.90, that a 0.95 CI for μ will have length no more than σ/4
Suppose that the continuous two-dimensional random variable (X, Y ) is uniformly distributed over the square whose vertices are (1, 0), (0, 1), (−1, 0), and (0, −1). Find the Correlation Coefficient ρxy
A company has 9000 arrivals of Internet traffic over a period of 18,050 thousandths of a minute. Let the random variable x represent the number of such Internet traffic arrivals in one thousandth of a minute. It appears that these Internet arrivals have a Poisson distribution. If we want to use the formula P(x)= (μ^x • e^−μ) / x! to find the probability of exactly 2 arrivals in one thousandth of a minute, what are the values of μ, x, and e that would be used in that formula?