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Suppose that 14, 10, 18, and 21 constitute a random sample of size 4 drawn from a uniform
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EBK AN INTRODUCTION TO MATHEMATICAL STA
- Suppose that X is a continuous unknown all of whose values are between -5 and 5 and whose PDF, denoted f, is given by f ( x ) = c ( 25 − x^2 ) , − 5 ≤ x ≤ 5 , and where c is a positive normalizing constant. What is the expected value of X^2?arrow_forwardSuppose that n observations are chosen at random from a continuous pdf fY(y). What is the probability that the last observation recorded will be the smallest number in the sample? I asked this question earlier today, but didn't quite understand all of the response. P(y1<=yn)p(y2<=yn) and so on was used, but shouldn't the yn be listed first in the inequality since we want to know if yn is the smallest?arrow_forward1. Consider the Gaussian distribution N (m, σ2).(a) Show that the pdf integrates to 1.(b) Show that the mean is m and the variance is σ.arrow_forward
- Consider a random sample X1, … , Xn from the pdff (x; u) = .5(1 + (THETA)x) -1 <= x <= 1where -1 <= theta <= 1 (this distribution arises in particlephysics). Show that theta = 3X is an unbiased estimator oftheta. [Hint: First determine mu = E(X) = E(X).]arrow_forwardIf X1 and X2 constitute a random sample of size n = 2from an exponential population, find the efficiency of 2Y1relative to X, where Y1 is the first order statistic and 2Y1and X are both unbiased estimators of the parameterarrow_forward10 Suppose that X is a continuous random variable with pdf given by X(x) =cxα−1e−(x/β)α for x≥0, where α >0,β >0 are some parameters of the distribution, and c is a constant. (a) Find the value of c such that fX is a valid pdf.(b) Find the cdf and the quantile function of X.(c) Ifα= 2 andβ= 4, calculate P(X <1) and find the quartiles of X.arrow_forward
- Consider random variables Xand Y with following joint pdf given as f(x,y) ={x+y 0≤x≤1, 0≤y≤1, 0 elsewhere. Compute correlation coefficient, ρXYarrow_forwardLet X denote the temperature at which a certain chemical reaction takes place. Suppose that X has pdf f(x)={1/9(4-x^2) -1<=x<=2 0 otherwise a) Compute P(0≤ X ≤1)b) Obtain E(x) and Variance of Xarrow_forwardConsider a random variable Y with PDF Pr(Y=k)=pq^(k-1),k=1,2,3,4,5....compute for E(2Y)arrow_forward
- Consider a real random variable X with zero mean and variance σ2X . Suppose that wecannot directly observe X, but instead we can observe Yt := X + Wt, t ∈ [0, T ], where T > 0 and{Wt : t ∈ R} is a WSS process with zero mean and correlation function RW , uncorrelated with X.Further suppose that we use the following linear estimator to estimate X based on {Yt : t ∈ [0, T ]}:ˆXT =Z T0h(T − θ)Yθ dθ,i.e., we pass the process {Yt} through a causal LTI filter with impulse response h and sample theoutput at time T . We wish to design h to minimize the mean-squared error of the estimate.a. Use the orthogonality principle to write down a necessary and sufficient condition for theoptimal h. (The condition involves h, T , X, {Yt : t ∈ [0, T ]}, ˆXT , etc.)b. Use part a to derive a condition involving the optimal h that has the following form: for allτ ∈ [0, T ],a =Z T0h(θ)(b + c(τ − θ)) dθ,where a and b are constants and c is some function. (You must find a, b, and c in terms ofthe information…arrow_forwardSuppose that Y1, . . . , Yn is a random sample from a population whose density function isarrow_forwardAssume X_1, X_2, .....X_n are random samples from X~Exponential(θ).1) Find the MLE estimator of θ .arrow_forward
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