Let X ~ Exp(2). Find a real number a E [0, 1] such that the events {X € [0, 1]} and {X € [a, 2]} are independent.

Q: Let X1,.,X, be a random sample from Uniform(0,1). show that x converges in probability to µ.

A: # random variable Xi follow uniform distribution (0,1)  Mean(mu)=1/2  Var(xi)=1/12   i=1,2,3,.....n…

Q: - Lei x1, X2,.n be a random sample from N(u, o?), calculate the cstimators of the popuiation…

A: The following information has been given: Let x1,x2,...,xnbe a random sample taken from a normal…

Q: Let X be a continuous random variable with PDF, (ke* : fx(x) = : otherwise x>0.

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Q: Let X ~ Poisson(λ). Derive the moment generating function of X and use that to find E(X3)

A: Solution: Let X~Poisson(λ) The probability distribution of X is  P(x)=e-λλxx!    x=0,1,2,3,..

Q: let flx)=, V2 Cunsrder y=x? Find the probability densily funchon of Y

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Q: 5Find the Orthogonal trajectories of the family of r=aeº

A: Question:- Find the orthogorial trajectories of the family of r=aeθ ?Solution :- We have r=aeθ -(i)…

Q: .Prove that e- -dt < (1+ t²)* 1/4 6.

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Q: Exercise 37 You arrive at a bus stop at a random time. According to the bus schedule, a bus arrives…

A: poisson distribution

Q: Let X1, ., X, be a random sample from Uniform(0,1). show that i converges in probability to u.

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Q: Let X be a continuous random

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Q: Let TNN(u,6²). define a faction Fact):= 2t- a, t>a, and Fact):= K (a-), tsa Find the value of a that…

A: Given:

Q: Prove that cov(X, Y) = cov(Y, X) for both discreteand continuous random variables X and Y.

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Q: The probability that a soldier in particular region will be infected by a particular type of…

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Q: Suppose that we observe that X1, X2, . . . , Xn  are iid∼ U(0, 1). Show that X(1) converges in…

A: A sequence of random variables Xn is said to converge in probability, denoted as Xn→pX, if…

Q: converges weakly to x in a normed linear sp

A: Given that, xn converges weekly to x in a normed linear space X We have to show that Every…

Q: Let x~ exp (A) Show that T=s Xi is sufficiency Statistic.

A: Generally we use factorization theorem to obtain the sufficient statistic.

Q: Let X1, X2, ... , Xn be a random sample from N(μ, σ2). Find the Moment Generating Function of X̅.…

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Q: A class contains 30 students. The probability of failing in one of the courses for each student is…

A: Given information: The number of students is n = 30. The probability of failing in one of the…

Q: f fn → ƒ uniformly and the functions fn have only jump discon- sinuities, prove that f has only jump…

A: Here we use epsilon delta proof

Q: There is one male baby seal for every nine female baby seals. Let X be the number of male baby seals…

A: Let p=P(male baby seal) = 1/9 Here n=5 Note that X follows a Binomial distribution with n=5 and…

Q: Assume that a population of patients contains 30% of individuals who suffer from a certain fatal…

A: Product decomposition:- For a group of random variables, the joint distribution can be decomposed as…

Q: How do i find e(x), var(x) and e(x^3) using the moment generating function

A: Let us define the random variable X with moment generating function Mx(t).The moment generating…

Q: 6. If X is t-distributed, show that X² is F-distributed.

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Q: Suppose that X is a continuous random variable whose pdf satis fx(a) = 0 for all a 2. Prove that…

A: From the given information, the random variable X is a continuous random variable whose pdf…

Q: Using a= 0.01 to test the null hypothesis HO: B1 = B2 = 0, the critical F value Is %3D

A: Our aim is to find the F-critical value

Q: use the beta function to compute P(3<Y≤5).

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Q: Let X1, X2,... be a sequence of independent and identically distributed continuous random variables.…

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Q: If Σ An diverges then > can diverges O only if c>0 O only if c< 0 O only if c+0 O for any real value…

A: Given, the series ∑an diverges, then we have to find the values of c for which the series ∑can…

Q: Suppose X1, X2, and X3 are independent and N(0.1)-distributed. Com- pute the moment generating…

A: Given function Y= X1X2+X1X3+X2X3 We have to calculate moment generating function

Q: The numbers x and y are chosen at random from a circle of radius 6. What is the probability that x +…

A: Probability = Area under interest/ total area Where Area under interest = shaded area Total area =…

Q: dz < 37 , 71,3

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Q: Let fn(x) x + n a. Show that fn(x) converges pointwise [0, 0). on

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Q: Let {Xn} and {Yn} be sequences of random variables such that Xn diverges to o in probability and Yn…

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Q: LetXbea randomvariable havingthe uniformdistribution ontheinterval(θ,θ+1),θ∈R.Showthat Xis not…

A:

Q: A class contains 50 students. The probability of failing in one of the courses for each student is…

A: From the given information, There are 50 students. The probability of failing in one of the courses…

Q: The autocorrelation function of a random process Rx (T) cannot have any arbitrary shape.

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Q: A class contains 30 students. The probability of failing in one of the courses for each student is…

A: From the given information, There are 30 students. The probability of failing in one of the courses…

Q: In an area having sandy soil, 60 small trees of a certain type were planted, and another 60 trees…

A: Let  x= No. of trees planted in sandy soil y= No. of trees planted in clay soil Px=0.7,…

Q: Prove that if X1, X2, . .. , Xn are i.i.d. Exponential(A) random vari- ables then X1 + X2 + ..+ Xn…

A: Given that X1 ,X2, X3 ,...., Xn follow i.i.d exponential distribution with parameter λ. It is…

Q: In an area having sandy soil, 40 small trees of a certain type were planted, and another 40 trees…

A:

Q: a Find the Orthogonal trajectories of the family of r =; 1+coso

A: we need to find the orthogonal trajectories of the given family of the curve in the polar form.

Q: Let Y be a random variable with moment-generating function m(t) = }e=t + } + je4t, where -00 <t<∞.…

A: Part (b) The expression for E(Y2) is, E(Y2)=d2M(t)dt2t=0 The differentiate the equation twice with…

Q: Let X denote the length (in seconds) of the next smile of a ran- domly selected 8-week old baby.…

A: It is a problem of a uniform probability distribution.

Q: The average number of customers at a windów of a certain bank per minute during banking hours is…

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Q: 5. Show that |Logz|<|In(z)|+ t, for all z # 0.

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Q: Let X be a random variable with cumulative distribution function 3 x >0 F(x) = x < 0 Determine its…

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Q: Let X be a random variable with p.d.f f(x) =x(9-x), 0sxS 3. Find the mode.

A: Introduction Mode: Mode is one of the types of measures of central tendency. Mode is the location…

Q: Let {Xn} and {Yn} be sequences of random variables such that Xn diverges to ∞ in probability and Yn…

A: