Date:itis knon that the random variable X withthe probability mass punction is as Follows:YA:() ; for X = 0,1,2ン>○P(x):: For othes xa) Determine the moment generating Function forthe sandom variable xX 50

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Answered: Date: it is known that the random…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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4. Two independent random variables x₁ and x₂ are both uniformly y₁=X₁X2/(x₁+x2). Hint: distributed between 9 and 11. Find the pdf of introduce the dummy variable y2=X2 and use the approach of two functions of two variables.
6. An optical device is used to detect the passage of cars in a single lane of a downtown street. Because there must be at least half a second between successive cars, it is assumed that the times T; between = 0.50 + Si, where S1, S2, ··. are independent exponential (A) random cars are of the form T; variables. (a) Find the mean and variance of each T;. (b) Let Y, be the time at which the nth car passes the detector. Calculate the mean and variance of Yn. (c) Under what conditions is Yn approximately normally distributed and why? (d) When n = 50 and A = 0.10, calculate the approximate probability that Yn exceeds 500 seconds.
b) Suppose that a continuous random variable X is associated with the following function: (i) Show that c = =1/15 2 71 Page 0 < x < 2, Scx, f(x) = {0, elsewhere. lo, (ii) Calculate the mean and variance of X. (iii) Find the CDF of the random variable X, F(x). (iv) Use either the probability distribution function (pdf) of X, f(x), or the cumulative probab distribution function (cdf) of X, F(x), to compute P(0.25 < x < 1.25).
Solve c,d
d) The random variable X has a Poisson distribution such that P(X= 0) = P(X = 1). Calculate P(X = 2).
Suppose X is a random variable of uniform distribution between 1 and 7. Find E(X)
For E (0,1) let Xp be a Geometric random variable with parameter p. (a) Find a value of p so that P(Xp> 2.5) = 9. (b) Let An be the event that Xp is even. Determine P(A,) in terms of p. (c) Suppose Y, is a random variable which is equal to the remainder after integer division of X, by 3. Let p = i, and determine the conditional probability mass function of conditioned on the event Yı = 1.
9. Given that f(x, y) = (2x+2y)/2k if x = 0,1 and y = 1,4, is a joint probability distribution function for the random variables X and Y. Find: (f(x|y = 1)
2. Suppose that X1, . Xn are iid Geometric random variables with frequency func- ... tion f(x; 0) = 0(1 – 0)", x = 0, 1, 2, ..., 0 E (0, 1). Find the ML estimator 0, of 0. Show that Ôn is consistent an find its asymptotic distribution.
4.2 The probability distribution of the discrete random variable X is f(x) = 3 (*)*** Find the mean of X. 3-x (¹)ª (²³)³—ª, x = 0, 1, 2, 3.
Bapu
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Date: it is known that the random variable X with the probability mass punctron is as Follows: Y4:(&) ; for X = 0,1,2 P(x): : For othes x a) Determine the moment generating Function for the sando variable x. X50