Let Y ∼N(μ,1) for μ unknown. Use the fact that P(|Y −μ|< 1.96σ) ≈0.95 to construct a random interval (a(Y ),b(Y )) (that is, an interval whose endpoints are r.v.s), such that the probability that μ is in the interval is approximately 0.95. This interval is called a confidence interval for μ; such intervals are often desired in statistics when estimating unknown parameters based on data.

Q: Consider a random sample X1, . . . , Xn from the uniform distribution U(−θ, θ) with parameter θ > 0…

A: *Answer:

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: 2. Let X have the N(u, o2) distribution. Find the MGF of X by using the fact that X = oZ+u where Z…

A: Given that, The probability density function of the normal distribution with parameter  and  is…

Q: Let X and Y have a bivariate normal distribution with parameters μ1 =20, μ2 = 40, σ21 = 9, σ22 = 4,…

A: Given: μ1=20μ2=40σ12=9σ22=4ρ=0.6 The conditional distribution of Y-X is Nμ1+ρσ2σ1x-μ, σ22(1-ρ2

Q: Let Y be a continuous random variable having the standard beta distribution with parameters a = 3…

A:

Q: B2. Let A₁,..., An> 0 and let X₁,..., Xn be independent random variables with common mean E(X) = μ…

A: This question deals with the minimum variance of the estimator μ^(X1,X2,...,Xn) for the mean μ given…

Q: (a) Show that T₁ is an unbiased estimator of 1/A. (b) Find the constant a such that T₂ = aX(1) is an…

A: It is given that  X1, X2, ....Xn follows Exponential with mean 1/λ then the variance of X is 1/λ2…

Q: For an SRS {Y: 3;=, from the normal distribution. Si= 1 fy (y;u, 0²) = ==// (Y-M) 2² 2 e √22√52 Show…

A:

Q: Let Y₁, Y2 , . . . , Y₁ denote a random sample from a Poisson distribution with mean X and define Wn…

A: Let denote a random sample from a Poisson distribution with mean and define Wn as:

Q: Suppose X has a gamma distribution with the following parameters: mean u- 41.5692 ms B = o /u =…

A:

Q: IQ scores are normally distributed with μ=100 and σ=15. A simple random sample of n=30 students is…

A: Given P(x&gt;110)≈ .0001 which means Only about 1 out of 1,000 samples will have a mean IQ larger…

Q: Let X be a discrete random variable with the following PMF: for x = 0 Px(x) = = 12 L3 L6 for x = 1…

A: Step 1:The probability mass function is given asx012P(x)1/21/31/6The range of the given PMF would be…

Q: 2. Given f(x) is the pdf of a discrete random variable x: a. f(x) = 6-x 18 P(4 < x≤ 10) = x = 0 x =…

A:

Q: If X is b(100,0.1), how do you find the value of P(12 < X < 14) using the normal approximation and…

A: The given data is as follows:The probability shows how likely an event occurs and its value lies…

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: Let x be a random variable which approximately follows a normal distribution with mean μ = 0.01 and…

A: Normal Distribution :  It also referred to as gaussian distribution .  Normal probability is a type…

Q: Use what you know about order statistics to show that for the random sample of size n = 3 the median…

A:

Q: Consider a random sample X₁, X2,..., Xn from the pdf f(x) = (1+0x) -1≤x≤1 where -1 ≤ 0 <1 (this…

A: f(x)=1/2(1+θx); -1≤x≤1