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Find the squared-error loss
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EBK AN INTRODUCTION TO MATHEMATICAL STA
- Show that the mean of a random sample of size n from an exponential population is a minimum variance unbi-ased estimator of the parameter θ.arrow_forwardLet X be a continuous random variable with pdfarrow_forwardLet Y1, Y2, ... , Yn be a random sample of size n from a gamma distribution with parameters α = 1and β = 2. Derive the probability distribution of the sample mean Y̅ using moment-generatingfunctions.arrow_forward
- The combined amount of time three customers spend waiting to receive service at a drive-through facility is a Gamma distributed random variable with parameter theta = 0.4 per minute. What is the probability that 3 randomly selected customers will have a combined waiting time of at least 10 minutes?arrow_forwardConsider random variables X1, · · · Xn, independent and identically distributed such that each Xi ∼N (0, 1). Write down an expression for the joint pdf of the n-dimensional random vector (X1, · · · Xn).(i.e. what is the distribution of a random sample of n standard normal random variablesarrow_forwardWhat is the Marginal PDF of f(x)=2? Is the event independent?arrow_forward
- Let y1,y2,...,y10 be a random sample from an exponential pdf with unknown parameter λ. Find the form of the Generalized Likelihood Ratio Test for H0: λ = λ0 versus H1: λ doesn't equal λ0. What integral would have to be evaluated to determine the critical value if α were equal to 0.05?arrow_forwardSuppose that X has the uniform distribution on the interval [0, 1]. Compute the variance of X.arrow_forwardSuppose the random variable y is a function of several independent random variables, say x1,x2,...,xn. On first order approximation, which of the following is TRUE in general?arrow_forward
- Without using a moment generating function; Prove that the variance of a beta-distributed random variable with parameters α and β is σ2 = αβ/[(α + β)^2 (α + β + 1)]arrow_forwarda. What is the probability that the lifetime X of the first component exceeds 3? b. What are the marginal pdf's of X and Y? Are the two lifetimes independent? x. What is the probability that the lifetime of at least one component exceeds 3?arrow_forwardSuppose the distribution of the time $X$ (in hours) spent by students at a certain university on a particular project is gamma with parameters $\alpha=50$ and $\beta=2 .$ Because $\alpha$ is large, it can be shown that $X$ has approximately a normal distribution. Use this fact to compute the approximate probability that a randomly selected student spends at most 125 hours on the project.arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage