7. In bacterial counts with a haemacytometer, the number of bacteria per quadrat has a Pois-son distribution with probability mass function f(x), where f(x) = 0" e- /x! and 0 is to beestimated. If there are many bacteria in a quadrat, it is difficult to count them all, and sothe only information recorded is that the number of bacteria exceeds a certain limit c, a largepositive integer. In a random sample of n quadrats, it was observed that ng quadrats hadx bacteria (x = 0,1, ., c) and that m quadrats had more than c bacteria. Show that themaximum likelihood estimate of 0 is given by-1mf(c)1- F(c)]= s In – m -where s =Ecn, and F(c) =E{(x).x=1x=0

Solution: 7. From the given information,

Answered: In bacterial counts with a…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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7. If a random variable has the probability density function -1≤x≤3 otherwise f(x)=k(x²-1) =0
A computer can provide you an infinite sequence of random integer numbers between 1 and 5 with the probability mass function P(X=x)=1/5 for x ∈ {1,2,3,4,5}.Using only this random sequence provided by the computer, generate a random sequence of integer numbers between 1 and 7 with the probability mass functionP(Y=y)=1/7 for y ∈ {1,2,3,4,5,6,7}.
Suppose the random variable, X, follows a geometric distribution with parameter 0 (0 < 0 < 1). Let X₁, X2,..., X be a random sample of size n from the population of X. (a) Write down the likelihood function of the parameter. (b) Show that the log likelihood function of depends on the sample only through Σj=1 Xj. (c) Find the maximum likelihood estimator (mle) of 0. (d) Find the method of moments estimator (mme) of 0.
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Jobs arriving to a computer have been found to require CPU time that can be modelled by an exponential distribution with parameter 1/140 per millisecond. The CPU scheduling discipline is quantum-oriented so that jobs not completing within a quantum of 100 milliseconds will be routed back to the tail of the queue of waiting jobs. Find the probability that an arriving job will be forced to wait for a second quantum. Of the 800 jobs coming in during a day, how many are expected to finish within the first quantum.
If λ = 6 per hour and μ = 6 per hour in an (M/M/1): (5/FIFO) model, find the probability that there is no customer in the system.
b) Sketch the probability mass function and the cumulative distribution function of X (c) Consider a random sample of 6 parcels. If Y is the number of parcels taking longer than 1 day to arrive, write down the distribution of Y and its mean and variance.
Suppose that the random variables X and Y have the following joint probability density function. f(x, y) = ce-8x - 5y, 0 0).
A catalyst researcher states that the diameters, in microns, of the pores in a new product she has made have the exponential distribution with parameter λ = 0.25. Suppose that the diameters of 10 pores are independent. a) Find the probability that more than seven of them have diameters less than 3 microns. b) Find the probability that exactly one of them has a diameter greater than 11 microns.
According to some law in genetics, when certain types of peas are crossed, the probability that the plant yields a yellow pea is 0.75 and that it yields a green pea is 0.25. For a plant yielding 90 peas, find the approximate probability that at most 64 peas are yellow. Use the normal approximation with continuity correction.
Consider a nano ring consists of total 12 atoms as shown below. A black atom is initially placed at the outer rim of the ring as shown in the figure below. The black atom can randomly jump (with equal probability) from atom to atom along the outer side of the ring on either side as shown by the arrows. When the black atom passes by the grey atom, it emits a pulse of light with intensity I,. Compute the total average intensity of light emitted by the black atom in 15 jumps.

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