Exercise 6. Let 0 > 0 and X ~U[0,0], i.e. X is uniformly distributed on the interval [0, 0]. a) As a function of 0, determine P(X < 1). b) Assume that 0 is unknown, but we can observe X. For given 00, we want to test the hypothesis Ho: 0 > 0o against the alternative H1: 0 < 0. Consider the test which rejects Ho, if and only if X < c. How should we choose c, as a function of 0o and a, to get a test with significance level a? Carefully justify your answer.

Q: Suppose that X₁,..., Xn is a random sample from a distribution with pdf f(x) = 2x, TE (0, 1). (a)…

A: Given that X1, X2,..., Xn is a random sample from a distribution with pdf f(x)=2x, for x∈0, 1.

Q: Let Y₁, Y2,..., Yn denote a random sample of size n from a population with a uniform distribution on…

A: To find the expected value of the smallest order statistic Y(1) from a uniform distribution on the…

Q: Let X1,...,X, be a random sample from (1+ 0x) f(x) = 2 ,-1< x < 1, -1 < 0 < 1. Find the consistent…

A: The random sample consists of X1,X2,...,Xn. Need to obtain: The consistent estimator of θ. The…

Q: Suppose that a pdf for a continuous random variable Y takes the form ayya-1 (1 + yª)r+1 y > 0 f() =…

A:

Q: (1)(A). (n)" Let Y₁, Y₂Yn denote a random sample of size n from a population with a uniform…

A: Given, Y1, Y2,......Yn denote a random sample of size n from a population with a…

Q: Let X₁,, Xn be a random sample from a population with probability mass function (pmf) p(x) = 0 (1 -…

A: Solution

Q: Let X1 be the sample mean of a random sample of size n from a Norinal(µ, o†) population and X2 be…

A: A statistic t is said to be an unbiased estimate of a parameter θ if Et=θ. Given μ^=ω X¯1+1-ωX¯2…

Q: Let X₁, Xn be a random sample of size n from the U(0, 0) distribution, where > 0 is an unknown…

A: Given that  be a random sample of size  from the  distribution, where  is an unknown parameter.

Q: 27. If X is uniformly distributed over (0, 1) and Y is exponentially distributed 1, find the…

A: X is uniformly distributed over (0, 1) and Y is exponentially distributed with parameter .

Q: Let U1, U2, U3, .,U48 be independent and identically distributed U,~Unif(0,1) and X = U, + U2 + U3 +…

A:

Q: If X has a uniform distribution U(0, 1), find the pdf of Y = ex with ndf

A:

Q: Establish a formula similar to (3.3.18. for the distribution. If X~ gamma (x,B) gamma then for any…

A: X is a Gamma random variable,γ(α,β) with PDF given as:

Q: 6. (For stat 5504 only). Let Y = X² +XZ where X and Z are independent, normally distributed and E(Z)…

A: Given :E(Z)=E(X)=2 and E(Z2)=E(X2)=3a) The predictor of Y from X. if X=1.8.b) The best linear…

Q: Use the method of moment to estimate the parameter O for the based on a random sample Xi, X2, a…

A:

Q: 5. Let X₁, X2,..., Xn are random samples from the pdf f(x; 0) = 0/x², 0 < 0≤x. a) Find the mle of 0.…

A: f(x;θ)=θx2;0&lt;θ&lt;x

Q: Let Z~ N(0, 1). Find P( - 2.4 < Z < 0.9). Submit Question

A: Solution-: In this case, Z is the standard normal random variable with μ=0 and σ=1. Mathematically,…

Q: Q4) a) Let , and 0, be two unbiased estimators of 0. Show that ô, = aô, +(1– a)ê,, 0sasl is an…

A: Q 4a) Here it is given that θ1^ and  θ2^are two unbaised estimater of θie E( θ1^)=E( θ2^)=θNow Here…

Q: 2. Let X and Y be two continuous random variables with joint pdf fxy(x,y). Let U = XY. Prove that…

A:

Q: f X1,X2,...,Xn constitute a random sample of size n from a geometric population, show that Y = X1 +…

A: Given information: Let x1,x2,....,xn  is a random sample of size n from a geometric distribution.…

Q: X and Y are continuous random variables with pdf f(x,y) = 2x for 0 ≤x ≤y ≤1, and f(x,y) = 0…

A:

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.

Q: Let X1, . . . , Xn be iid with pdf f(x) = 1 x √ 2πθ2 e − (log(x)−θ1) 2 2θ2 , −∞ < x < ∞, and unknown…

A: Considering that  X1,X2,.........,Xn are iid with the following density as given.

Q: Let X₁, X2, ..., X and Y₁, Y2, ..., Ym be random samples from populations with moment gener 25…

A: Given information: X1, X2,....Xn and Y1,Y2,....Ym be the random samples from the population with…

Q: Let X1,., Xn(7n > 2) be independent random variables with pdfs S(1,10) = { , if -i(0 – 1) 0. Find a…

A:

Q: Let X₁, X2, ..., X and Y₁, Y₂, ..., Ym be random samples from populations with moment generating 25…

A: Given X1, X2, . . ., Xn and Y1, Y2, . . ., Ym be random samples from populations with moment…

Q: Find the mean and standard deviation for each uniform continuous model U(1, 13) U(70, 200) U(3, 92)

A: Obtain the mean and standard deviation for the uniform continuous model U(1,13). Obtain the mean of…

Q: Exercise 4. Let X and Y be jointly continuous random variables with joint PDF given by ƒx,y(x, y) =…

A: Solution

Q: Find the P-value.

A: Given : n=1000 , X=850 , p0=0.9 , α=0.063 The estimate of the sample proportion is ,…

Q: Let X₁,..., Xn be a random sample of size n from the U(0, 0) distribution, where > 0 is an unknown…

A: Given that  be a random sample of size  from the  distribution, where  is an unknown parameter.

Q: Suppose X and Y are continuous random variables such that the pdf is f(x,y) = x + y with 0 < x < 1,0…

A: Given information: The joint probability density function of two continuous random variables X and Y…

Q: b) Let Z₁: = X-X~N(0,1), and W₁=Y-Y~N(0,1), for i = 1,2,3,...,10, then: oy i) State, with…

A: Given Zi~N(0,1) and Wi~N(0,1)  Result: t~F(a,b) Then  Mean=E(t)=b/(b-2)…

Q: Suppose X and Y are independent random variables, where X is uniform on (0,1), and Y is exponential…

A: According to the given information, X~U0,1 Y~exp3

Q: Let Y₁, Y2, ..., Yn be a random sample from a population that is uniformly distributed on the…

A: Yi ~ U(θ,5θ) ; i=1,2,...,n

Q: Suppose that that the RV X has the Gamma(2,β ) distribution for some β > = 0, E(X) = 2β and V (X) =…

A: Given that the RV X has the  distribution for some ,  and .Given , the RV Y has the  distribution.

Q: Let X1, . . . , Xn, . . . ∼ iid Bern(θ). Consider the Bayes estimator under squared error loss with…

A: The Bayes estimator under squared error loss for the parameter  given a uniform (0,1) prior is the…

Q: 2. A random variable X has the exponent ial distribution as f(x) = de¯^", x > 0 %3D a) Find the mom…

A: Given   X~ exp(lambda)  We want to find mgf and mean variance through mgf

Q: 2) The reading on a voltage meter connected to a test circuit is uniformly distributed over the…

A: 2.a)The random variable Y defines the reading on a voltage meter connected to a test circuit.

Q: Let X1, X2, X~N(0, 1), find maximum likelihood estimator of 0. x = 0,1,2,... If X-f(x)= e.w. find…

A: Here given that X1,X2,....Xn follows the normal distribution with parameters θ and 1.Then, we have…