We'll say that a permutation T = (T(1),..., T(n)) contains a swap if there exist i, j E {1,.,n} sothat T(i) = j and A(j) =i, where i + j... .(a) What is the expected number of swaps in a random permutation?(b) What about the variance?(c) In the same spirit as above, we'll say that T contains a s-cycle if there exist i1,..., is E {1,...,n}with T (i1) = i2, T (i2) = i3,..., T(i,) =i1. Compute the expectation of the number of s-cycles.•...

a) Let we take n > 1. Let Xi be a random variable if i is a partner in a swap or not. The values…

Answered: We’ll say that a permutation T =…

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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show that if X11, X12,..., X1, X21, X22,..., X2n₂ are independent random variables, with the first n constitut- ing a random sample from an infinite population with the mean μ₁ and the variance of and the other n2 constitut- ing a random sample from an infinite population with the mean μ2 and the variance o2, then 1 M2 (a) E(X₁-X₂) = μ₁ −μ2; 07 0 22 (b) var(X₁-X₂) = + n₁ n₂
1. The random variables §, §1, §2,... are independent and identically distributed with 1/4 and P(§ = j) = c/j for j = 1,2,3. Let X₁ = 0 and Xn = max(§₁, ..., Èn) for n = 1, 2,.... distribution P(§ = 0) = (a) What value must c take? (b) Explain why {Xn, n = 0, 1, 2,...} is a Markov chain. (c) Write down the transition matrix. (d) Draw the transition diagram and classify the states (aperiodic, transient, re- current, eorgodic, etc). (e) Calculate P(Xn = 0). (f) Calculate P(X₁ = 3, X₂ = 1|X₁ = 3).
2. Some properties of Expected value and variance of a random variable. a) Assume that X is an arbitrary discrete random variable, and a and b are constant. Using the definitic Show: and E(aX + b) = a · E(X) + b V(aX + b) = a² · V(X ) Stat 3128 Ali Mahzarnia P STAT 3128 Ali Mahzarnia b) Justify the computational formula of Variance of a random variable which is to justify : V(X) = E[(X – µ°] = Ex – µ)P • ptx) = | 2: Here needs justification By Cauchy Schwarz inequality it can be shown that the right hand side is always positive. Analogs expression in Mechanic : Parallel axis theorem Iem = I– md² Moment of Moment of inetria about Inertia of an an axis shifted object about the center of a mass by d from center of mass (a parallel shift) Icm is dispersion around the mean and is like second central moment (variance) I is like second moment if d is mean m is like sum total all the weight of each of the x which all add up to 1 d squared is like squared of mean since we,
3. An integer is randomly chosen from the set 1, 2,..., 100. If this integer is divisible by 2 we let Y = 0, and if it is not divisible by 2 but is divisible by 3 then Y = 1. We let Y = 2 in all other cases. Find mean and variance of Y.
Assume A and B are independent events with P(A) = 0.14 and P(B) = 0.55. Find (a) P(A N B), (b) P(A U B), (c) P (A'n B'), (d) P (A' N B), and (e) P (ANB'). ..... (a) P(A N B) = (Type an integer or a decimal.) (b) P(A U B) = (Type an integer or a decimal.) (c) P (A' n B') = (Type an integer or a decimal.) (d) P (A'N B) = (Type an integer or a decimal.) (e) P (AN B') = (Type an integer or a decimal.)
Example 2.8. Suppose X is a random variable whose CDF is: 3 Fx(z) = lp1.00)(2) + 100.0)(2) +20.00)(2) + Ig0,(?) Determine the following: a) fx(1) b) fx(0.1) c) fx(100) d) P(х > 105) e) P(0 0.1)

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