2.5. Let N be a random variable with distribution function 0 if <0 F(2) = 〈a,2 if 0 < x < 1 Find P{1,4 < X < 5), P12 < X < .8}, and P(X = 1/2).
Q: 10. Let X1, X2, .. , Xn be a sequence of n random variables where with E(X,)=µ for each i. What is…
A: Solution: 10. From the given information, X1, X2,………….,Xn are n random variables and the E(Xi)=µ
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A: Solution-: Let, X -be a continuously random variable takes values between 0 and c i.e. P( 0≤X≤c)=1…
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Q: (34) Let X be a random variable with p.d.f. 1<x<» k f(x) = { find k. O.w
A: Formula used: fx=x-2k 1<x<∞0 Otherwise
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A: The PMF of X is given by, P(X=x) = e-λ λxx! , x=0,1,2,.. X1, X2,...,Xn are iid collection of random…
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A: The pdf is given as,A random variable is also defined here.The objective is to compute the variance…
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Q: 6. Two independent random variables X1=(-1,0,1) and X2=(-1.0) can take the following probability…
A: Expected value is also known as mean or average. It used to take average for discrete or continuous…
Q: 5. Let Y1 01, - 0 0. 02 02 (a) Find the joint pdf of Z1 = Y1, Z2 = Y2 and Z3 = Y1 + Y2 + Y3. (b)…
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Q: O Find E[S„] and Var[S„].
A: It is given that Xis are iid and follows Uniform distribution with parameter a and b.
Q: 2. Let X be a random variable with the Poisson distribution, parameter 2. Show that, for w = 1, 2,…
A: X~Poisson(λ) Therefore P(X=x)=e-λλxx! ; x=0,1,2,……
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A: Solution
Q: (42) Let X be a random variable with p.d.f. ke 0<x<0 f(x) ={ find k. O.W
A: Given : f(x)=ke-23x 0<x<∞
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A: Option (I) is correct as E(X2)>= (E(X))2 when we find Variance.
Q: (40) Let X be a random variable with p.d.f. (k +4)e¯3x f(x) ={ 01) O.w
A: a)Use the condition of valid pdf to find the value of k…
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Q: (5) Let X be a random variable with p.d.f. k 1<x<o 2 f(x) =< , find k. O.w
A: Given: fx=kx2 1<x<∞0 Otherwise
Q: 6. [Maximum mark: 5] The following table shows the probability distribution of a discrete random…
A: Thank you 😊
Q: n F. De X(w) i- -X(w)
A: the distribution function is also known as a cumulative distribution function.shows that the…
Q: Q 6.1. Suppose Z = (Z1, Z2, Z3) is a standard multi-variate Gaussian random variable i.e., for i≤ 3,…
A: Given that be a standard multi-variate Gaussian random variable.For are i.i.d. random variables.
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A: Let X1, X2,.......Xn be the random sample having Uniform distribution (0,1)Y1=min{X1,…
Q: 20) Let X be a random variable with p.d.f. k T<x<10 f(x) =- , find k. O.w
A: Solution: Let X be the random variable with pdf f(x) is f(x)=kx 1<x<100 otherwise
Q: 48) Let X be a random variable with p.d.f. 1 -8x 0<x<00 k f(x)= , find k. O.w
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Q: 8. Let X1, X2 be i.i.d. random variables from a distribution with p.d.f xe r>0 fx(x) = otherwise and…
A: The distribution is given by, fX(x) = xe-xx>00Otherwise So, P(X≥x)=∫x∞fX(t) dt=∫x∞te-t…
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A: ANSWER: For the given data,
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Q: Let X1, X2, ..., Xm be a random sample of size m from a Binomial distribution with p = 0.25 and n…
A: Given: p=0.25 4, 6, 4, 6, 2
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- 7. Let Y₁,...,, denote a random sample from the uniform distribution on the interval (0,0+1). Let: 3₁ = 8 - 12/2 Y 8₂ = Y(n) n n+1 Show that both ₁ and 2 are unbiased estimators of 0. >-{ = [1 0I1. The probability distribution function, P(X-x) of a discrete random variable X is defined as for =0 P(X -x) 4. for x=1 for =2 a) Find the cumulative distribution of X, Fx) b). Find i) P(X-1) ii) P(0(8) Let X be a random variable with p.d.f. 1The random variables §, 1, $2,... are independent and identically distributed with distribution P(§ = 0) = 1/4 and P(§ = j) = c/j for j = 1,2,3. Let X₁ = 0 and Xn = max(§₁,..., En) for n = 1, 2, .... (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(X4 = 3, X₂ = 1|X₁ = 3).(40) Let X be a random variable with p.d.f. (k +4)e-3x 01) O.w10. Let X be a continuous random variable that only takes on non-negative values. Suppose E(X)=4 and V(X) = 1, which of the following statements are necessarily true? O Pr(2 { Pr(X > 8) { 3 4 O Pr(X > 10) < 16A random walk (RW) {Sn}n>o is a sum of id r.v.s X1, X2, •• , Xn,withP(X1 = a) = p, P(X1 = b) = 1 -p= q.Find the expectation E(Snl and variance var(Sn) of Sn for any n.Question 4 Let X be a random variable with AX) 0, EX) 0.6. if x 1 Answer (1) and (2): (1) If X, X2, X3, X4 are iid f, then (i) PX4) S0.8) = (.. ..). (ii) PX1) < 0) = (...). (2) Let X, X2,.., X36 be iid f You are given: PZS 1.96) = 0.975; P(ZS 2) = 0.977; P(Z< 1.55) 0.939, PZS 1.81) = 0.965. Use the CLT to approximate PXS 0.2). %3D %3D %3DLet X₁, X₂ be 2 mutually independent discrete random variables. The first variable X₁ can be a whole number from 1 to 5. Its distribution function is 6 - i m₁ (i) 15 The second variable X₂ can be a whole number from 1 to 6. It has a uniform distribution. First, find the exact values (using fractions) of P(X₁ = 3) = 1/5 : P(X₂ = 5) = 1/6 P(X₁ = 3 and 3 and X₂ = 5) = 1/30 X₂ # Let Y denote the maximum of the X¿'s. Find the probability that Y = 1: P(Y = 1) = 1 Find the probability that Y = 2, P(Y= 2) = 2/45 Hint: Use cases based on what X₁ and X₂ are.2. Let X1, X2, ..., X, be iid random variables with a common pdf/pf f(x, 8) depending on an unknown parameter 0. Using the factorisation criterion show that any one-to-one function of a sufficient statistic is a sufficient statistic.2) Let X₁, X2, X3, X4 be a random sample of size 4 from a population with the following distribution function Where, ß > 0. 1 f(x; 0) = B 0, e (x-4) B for x > 4 otherwise f) Is the maximum likelihood estimator of the parameter ß unbiased or not? g) What is the maximum likelihood estimator of g (B) = 2(B + 1). h) Is the maximum likelihood estimator of g (B) = 2(ß + 1) unbiased or not?(1) Let X be a random variable with p.d.f. 1-8x 0SEE MORE QUESTIONSRecommended textbooks for youA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSONA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON