4. Suppose X₁ and X₂ are independent random variables with cdf Fx (x) = sin(x), 0 ≤ x ≤ 1/ a. Show that fx(2) (x) = 2sin(x) cos(x), where X (2) is the random variable for the 2nd order statistic. (Remember, we only have 2 random variables.) . b. Show that fx) (r) = 2cos(r)[1 - sin(x)], where X(1) is the random variable for the 1st order statistic.

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A:

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A: Let  are iid with mean  and variance .Let

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A: Expectation gives the most likely value of a random variable.

Q: (a) Is EX; sufficient for 0? (b) Find a complete sufficient statistic for 0.

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Q: following (a) E(X) and E(Y) (b) Cov(X, Y) (c) Are X and Y independent? Justify your answer.

A: Solution

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A: The provided information is as follows:. and  are independent random variables..

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A: Note: Hi there! Thank you for posting the question. As there are several independent questions,…

Q: 29. Suppose X1, X2, X3 are a random sample of size 3 from a distribution with pdf f(x) = (1/0)ex/0…

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Q: 3. Let the random variable X have the pmf f(x) = = a) E(X) b) E(X²) c) E(3X²2X + 4) (x+1)² for x =…

A: The provided information is as follows:The probability mass function of  is .

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Q: 4. Let X₁, X2, Xn represent a random sample from a distribution with E(X) = 0. Consider two…

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Q: 7. Suppose random variables X and Y are independent. Let g(z) and h(y) be any bounded measurable…

A: The Given Random variables X and Y are independent. To show that CovgX,hY=0.

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A: Solution

Q: (1) E(E(X)) = E(X) (2) E(X – E(X)) = 0 (3) Var(X) = E(X²) – [E(X)]² (4) Given constants a and b,…

A: Discrete random variable is defined as a variable which can take only specific values. For example…

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A: E(S2) = (n-1/n)* sigma2  =! Sigma2  So it is biased estimator for sigma2

Q: (13) Let X be a random variable with p.d.f. 2e-2x 0<x<0 f(x) ={ , find E(e2*) O.W

A: Solution :  Given :  f(x) = 2e-2x    0&lt;x&lt;∞0   o.w     E( e-2x )  = ∫x e-2xf(x) dx…

Q: 2) Let X₁, X₂, X3, X4 be a random sample of size 4 from a population with the following distribution…

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Q: Suppose the conditional mean function is Y = Bo + B1X + B2X² + B3X3 + U where E[U|X]3D0. By mistake,…

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Q: 3. Suppose Y is a random variable with pdf f(y) = ay – b, 0 < y < 3 %3D 15 Find a and b given that…

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Q: 4. If X~ N3 (μ, E) where µ¹ = [2-3 1] and Σ = 1 3 2 | 1 2 2 then find the distribution of 3X1-2X2 +…

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Q: 4. If the random variable X has an show that for a > 0, b>0, exponential distrib f(x) = x/4, for x ≥…

A: For the random variable X~Exponential1λ, it is given that: fx=1λe-xλ, for x≥0

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Q: Let X₁, Xn be a random sample from the beta distribution with a = B = 0 and = = {0: 0= 1,2}. Show…

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Q: A statistic is denoted by T(X1,.., X„) or simply T(X) 14.- Let X1, X2 be a random sample from the…

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Q: 13. Find the MGF of a random variable whose moments are u, = (r +1)! 2". Ans Mx(t) = (1-2r) 14. If a…

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Q: (a) Find a sufficient statistic T(X₁,..., Xn) for p. (b) Show that 1 {X₁ = 1, X₂ = 0} is an unbiased…

A: It is given that X1, X2, ........, Xn follows Bernoulii distribution with parameter = p.

Q: 16 a) A random variable has pdf £x (x) = ² (1-x²) fx 4 0 (1-x²) 0≤x≤1 Find E (4x+2) and E(x²). else…

A: Given information: f(x)=541-x4      ; 0≤x≤10                     ; Elsewhere

Q: 3. The Cov(X) of a random vector X'= (X₁, ..., Xn) is defined as E{(X-E(X))'(X-E(X)}. True or false,…

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A: X and Z are independentV(X) = 4, Cov(X,Y) = 7

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Q: 2. If X is a random variable taking values 0,1,2,.... and P(X) = ab*, where a and b are positive…

A: i) First part:ii) second part:

Q: (7) Suppose that X₁ and X₂ are two random variables with j. m. g. f. 2 (t₂-1)(t₂-1) + (t₂-1)²' - ²₁…

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Q: 3.2.13 Y is a continuous random variable with { 2(1- y), 0< y < 1; fy(4) = { 0, otherwise. Derive…

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A: Note: Hi there! Thank you for posting the questions. As there are multiple sub parts, according to…

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A: Solution