1. Suppose that diseased trees are distributed randomly and uniformly throughout a large forest with an...
1. Suppose that diseased trees are distributed randomly and uniformly throughout a large forest with an average of λ per acre. Let X denote the number of diseased trees in a randomly chosen one-acre plot with range, 0,1,2,.. a) What distribution can we use to model X? Write down its probability mass function (b) Suppose that we observe the number of diseased trees on n randomly chosen one-acre parcels, X1, X2, ..., X The random variables Xi, X2, ...,X, can...
please help with all of these questions, thanks. Suppose that diseased trees are distributed randomly and uniformly throughout a large forest with an average of λ per acre. Let X denote the number of diseased trees in a randomly chosen one-acre plot with range, 0, 1,2,...^ (a) What distribution can we use to model X? Write down its probability mass function (p.m.f (b) Suppose that we observe the number of diseased trees on n randomly chosen one-acre parcels, Xi, X2,...
6. For a certain section of a pine forest, the number X of diseased trees per acre has a Poisson distribution with mean λ = 5. (C) Suppose we have 4 plots of forest, each one acre in size, giving 4 independent Poisson random variables, X1, X2,..., X4 with mean λ = 5. Find the probability that at least two of the plots of forest have 2 or more diseased trees per acre.
4 Mary is a waitress in a city centre restaurant. She receives tips from customers at an average rate of λ per hour. During a particular eight hour shift, she receives 5 tips. We wish to estimate λ. a (2 marks) Let X be the number of tips Mary receives in an 8 hour period. If tips are received according to a Poisson process with rate A tips per hour, state the distribution of X, with parameters b (5 marks)...
Assume that we have three independent observations: where Xi ~ Binomial(n 7,p) for i E { 1.2.3). The value of p E (0, 1) is not known. When we have observations like this from different, independent ran- dom variables, we can find joint probabilities by multiplying together th ndividual probabilities. For example This should remind you the discussion on statistical independence of random variables that can be found in the course book (see page 22) Answer the following questions a...
Q2 (15%): Suppose that the number of items one purchased online last month Yi depended on one’s salary Xi follows a Poisson distribution Yi ∼ Poisson(β0 + β1Xi). We have n pairs of independent observations (x1, y1),(x2, y2), . . . ,(xn, yn) so that Yi are mutually independent. Please use Newton’s method to find the maximum likelihood estimation (MLE) for (β0, β1). Hint: Xis are fixed and Yis are random. Please first write out the probability mass function for...
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...
need to check my work. Just need B and C Problem 2. Recall that a discrete random variable X has Poisson distribution with parameter λ if the probability mass function of X is fx (x) = e-λ- XE(0, 1,2, ) ar! 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 cornputation that the mean of a Poisson randoln...
1. Suppose that y E R is a parameter, and {X1, X2, ..., Xm} is a set of positive i.i.d. random variables with density function fx, given by fx.(ar)yey, You observe that X = {X1, X2, ..., Xm} in fact take the values r = {r1, x2, ..., x'm}, respec- tively. Write for the average of the values {x1, x2,.., Tm) a) What is the likelihood function, L(y; x), as a function of y? What is the log-likelihood function, log...
4. Consider the following model: Let X = (1,X1,X2, Xs)". Suppose and that X is not perfectly colinear. Suppose further that ElY]oo and Exoo for 1 j S 3. Using a large sample of i.i.d. observations from (XX1.x2, Xs), you estimate this equation using Ols. Let β-(%, 33) de- note the resulting estimate of β = (Ah-…β3). (a) Let θ-2A-β3. Is θη-2A-A3 a consistent estimate of θ? Is it an unbiased estimate of 0? (b) Express Var(0,] in terms of...