5. When a muon decays, an electron is emitted at an angle 0 relative to the spin, such that theprobability density function of the angular distribution isdP = k(1+ax) dx,where x = cos 0 and a is a constant that must be determined experimentally. Find the valueof the constant k, and the mean and variance of the x distribution (in terms of a).

Find the constant kThe probability density function is given by:dP = k(1 + αx) dxSince this is a…

Answered: 5. When a muon decays, an electron is…

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...

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The Exponential pdf for continuous random variable Y takes the form e-ylß y > 0 f(V) = for ß > 0. y 1.6) Give your answer to three decimal places.
Q.2 Let X has the following probability density function: fx(x) = a e-ax;x > 0 Find moment generating function and then using it find mean and variance. End of Quiz
2. In probability, it is common to model the deviation of a day's temperature from the monthly average temperature using the Gaussian probability density function, 1.-/9 f(t) = V9T This means that the probability that the day's temperature will be between t = a and t = b different from the monthly average temperature is given by the area under the graph of y = f(t) between t a andt = b. A related function is %3D F(x) = -2/9 dt, 2 0. %3D V9n Jo This function gives the probability that the day's temperature is between t = -x and t = x different from the monthly average temperature. For example, F(1) 0.36 indicates that there's roughly a 36% chance that the day's temperature will be within 1 degree (between 1 degree less and 1 degree more) of the monthly average. 1 (a) Find a power series representation of F(x) (write down the power series using sigma notation). (b) Use your answer to (a) to find a series equal to the probability that the day's temperature will be within 2 degrees of the…
1. Develop a random variate (composition method) generator for the random variable X with the probability density function where I represents the indicator function. 5 f(x)=e*I(x 0) 4 a. Implement this generator to generate 500 random variates. b. Compute the sample mean and sample variance of these 500 variates and compare with the theoretical values. c. Plot the histogram of these 500 variates and compare the histogram with the graph of f(x).
Suppose that the interval between eruptions of a particular geyser can be modelled by an exponential distribution with an unknown parameter 0 > 0. The probability density function of this distribution is given by f(x; 0) = 0e 0¹, x > 0. The four most recent intervals between eruptions (in minutes) are x₁ = 32, x₂ = 10, x3 = 28, x4 = 60; their values are to be treated as a random sample from the exponential distribution. (a) Show that the likelihood of based on these data is given by L(0) 04-1306 = (b) Show that L'(0) is of the form L'(0) = 0³ e 1300 (4- 1300). (c) Show that the maximum likelihood estimate of 0 based on the data is ~ 0.0308 making your argument clear. (d) Explain in detail how the maximum likelihood estimate of that you have just obtained in part (c) relates to the maximum likelihood estimator of for an exponential distribution.
QUESTION 3 Suppose that the continuous random variables X and Y have the joint density function 4xy x, y = [0, 1] 0 otherwise f(x, y) = and suppose the expected values of Y is E(Y) Round to 4 decimal places if needed. = 3. Find the variance of Y.
Assume the time required to fully recharge an electric car is uniformly distributed between 160 and 220 minutes. What is the expected charge time, E(x)?
Let X be a random variable with exponential distribution having lambda=6 and let Y = 3X. a. Find P(X>0.25) b. Find the mgf of Y c. Find the pdf of Y
Solve both parts
2B. (a) An exponential distribution has probability density function ƒ(x) = ¹²e-x/a, x ≥0. α Show that the mean of the distribution is a and the variance is a². (b) A machine contains two belt drives, of different lengths. These have times to failure which are exponentially distributed, with mean times a and 2a. The machine will stop if either belt fails, and the failures of the belts are independent. Show that the chance that the machine is still operating after a time a from the start is e 32
Need sol of b part only. Within 20 minutes.

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