Ex. Let X be a Gamma distribution with parameter (a, A). 1 -re-d* (ar)ª- for 0<=x<∞. T(a) f(x) = Calculate M(t):
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A: Here, X follows uniform (0, 1). Consider Y=2x+5. The cdf of Y, as follows:
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Q: X follows a gamma distribution with PDF f(x) = 4xe-2x , where X > 0 (a) Derive E(Xn ).
A: Given, fx=4xe-2x, x>0 Therefore, Exn=∫0∞xnfxdx=∫0∞xn×4xe-2xdx=∫0∞4×xn+1e-2xdx=4×∫0∞xn+2-1e-2x dx
Q: Q5. A continuous r.v. X is said to have a Gamma (a, B) distribution if its moment generating…
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Q: [1,4] is 5. Find The average value of f on f(x) dx.
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Q: 5: Let f(x) := H(x) cos x, with 1, x> 0 0, x< 0. H(x) := Define the distribution T, by (Tf, ø) := L…
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A: Hello! As you have posted more than 3 sub parts, we are answering the first 3 sub-parts. In case…
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A: here given joint probability density function of z1 and z2
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Q: 1-cos(x) X has PDF f(x) = if 0 < x <T 0 otherwise %3D Find the probability P(0.38 < X < 3.12).
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A: It is known that E(Xi) =1/λ and Var(Xi) =1/( λ2) EY=EX1+X2+...+Xn =EX1+...+EXn…
Q: Suppose that L = 27, a2 = 1, and the initial temperature distribution is f(x) = 27 - x for 0 <x< 27.…
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Q: O = Let 0, X,, X2,... be RVs. Suppose that, conditional on 0, X1, X2,... are independent and X, is…
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Q: For a Gamma distribution with rate parameter λ=1 and shape parameter a = 3 ª f(x)=r(a) -xa-1e-ax x>0
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Q: Let X have a gamma distribution (a, ß)and Let Y = ex.Find G(y)and g(y) by D F method %3D
A: From the given information, X be a gamma distribution α,β. The pdf for X is, f(x)=βαΓ(α)xα-1e-βx The…
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Q: Let X follow the beta distribution with pdf S(x) = K z (1 – 2)² on 0<z< 1. Evaluate the value of K.
A: To find the value of K we will use the fact that the integration of f(x) over whole range of x is…
Q: Let X have a gamma distribution (a, ß)and Let Y = e*. Find G(y)and g(y) by D F method
A: From the given information, X be a gamma distribution (α,β).
Q: Ex. Let X be a Gamma distribution with parameter (a, 1). f(x). 1 -de-d*(ar)“-l for 0<=x<∞. I(a)…
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Q: Let X~N(0,1). Let Y=2X. Find the distribution of Y using the moment generating function technique.
A: Given Let X~N(0,1). Y=2X.
Q: Suppose that L = 27, a = 1, and the initial temperature distribution is f(x) = 27 - x for 0 <x < 27.…
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Q: w that X has an exponential distribution. State its paramete X.....X₁ are independent observations…
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Q: . Let Y,, Y2, .. , Y, denote a random sample from a population with pdf fyle) = (0 + 1)y®,0 -1 a)…
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- Recall that the general form of a logistic equation for a population is given by P(t)=c1+aebt , such that the initial population at time t=0 is P(0)=P0. Show algebraically that cP(t)P(t)=cP0P0ebt .X follows a gamma distribution with PDF f(x) = 4xe-2x , where X > 0(a) Derive E(Xn ).1. Let X have a gamma distribution with α > 1. Show thatE [1/X] = 1/[θ*(α −1)]
- If X is exponentially distributed with parameter λ and Y is uniformly distributed on the interval [a, b], what is the moment generating function of X + 2Y ?f X1,X2,...,Xn constitute a random sample of size n from a geometric population, show that Y = X1 + X2 + ···+ Xn is a sufficient estimator of the parameter θ.Let X1, .... Xn be a random sample from a population with location pdf f(x-Q). Show that the order statistics, T(X1, ...., Xn) = (X(1), ... X(n)) are a sufficient statistics for Q and no further reduction is possible?
- LetX1,X2,...,Xn be a sequence of independent and identically distributed random variables having the Exponential(λ) distribution,λ >0, fXi(x) ={λe−λx, x >0 0, otherwise Define the random variable Y=X1+X2+···+Xn. Find E(Y),Var(Y)and the moment generating function ofY.Q4) If X is a continuous random variable having pdf ke~ (2x+3y) x>0y>0 xy) = = e p(x) { 0 otherwise Find a) the constant k b) P(X>1) ¢) X, X2, 02, standard deviation.Does a distribution exist for which:Mx(t) = (t)/(t-1)for |t| < 1? If yes, find it, otherwise prove that is not possible
- Suppose X1, . . . , Xn are i.i.d. from a continuous distribution with p.d.f. fθ(x)=1/2(1+θx)if −1≤x≤1, where θ ∈ [−1, 1] is an unknown parameter. (a) Find E(X1).(b) Find the MME for θ. (c) Compute the variance of your MME from part (a).Let X1, . . . , Xn i.i.d. U([θ1, θ2]), i.e., X1, . . . , Xn are independent and follow a uniform distribution on the interval [θ1, θ2] for θ1, θ2 ∈ R and θ1 < θ2. Find an estimator for θ1 and θ2 using the method of moments.suppose x has an exponential distribution with probability density function f(x) =2e^-2x, x>0. Then P(X>1)
