6.82 If Y is a continuous random variable and m is the median of the distribution, then m is suchthat P(Y ≤m) = P(Y ≥ m) = 1/2. If Y₁, Y₂, ..., Y, are independent, exponentially dis-tributed random variables with mean ß and median m, Example 6.17 implies that Y(n) =max (Y₁, Y₂, ..., Yn) does not have an exponential distribution. Use the general form of Fy) (y)to show that P(Y(n) > m) = 1- (.5)".

It is given that Y1, Y2, …,Yn are independent random variables that follow an exponential…

Answered: If Y is a continuous random variable…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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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…
If X is a continuous random variable with X ∼ Uniform([0, 2]), what is E[X^3]?
.A sample of 9 measurements, randomly selected from a normally distributed population, resulted in x= 2.6, and s= 0.9 Conduct a hypothesis test to verify the claim that the population mean is greater than 2.5 . Use a=.05
In recent years, several companies have been formed to compete with AT&T in long-distance calls. All advertisethat their rates are lower than AT&T's. AT&T has responded by arguing that there will be no dierence in billingfor the average consumer.Suppose that the average bill for current AT&T customers is $21. A statistician takes a random sample of 100customers and recalculates their last month's AT&T bill using the rates quoted by a competitor. Let X be therandom variable for the amount of a recalculated bill, and µ be the unknown population mean of X. The samplemean and standard deviation for X are $20 and $6. The statistician wants to test if µ is signicantly dierent from$21, which is the average bill for current AT&T customers (d) Using 5% signicance level, calculate the critical values of the test. Is the null hypothesis rejected?(e) The competitor would react by using dierent set of hypotheses. For example, consider H0 : µ = 19 andH1 : µ 6= 19. What…
In recent years, several companies have been formed to compete with AT&T in long-distance calls. All advertisethat their rates are lower than AT&T's. AT&T has responded by arguing that there will be no dierence in billingfor the average consumer.Suppose that the average bill for current AT&T customers is $21. A statistician takes a random sample of 100customers and recalculates their last month's AT&T bill using the rates quoted by a competitor. Let X be therandom variable for the amount of a recalculated bill, and µ be the unknown population mean of X. The samplemean and standard deviation for X are $20 and $6. The statistician wants to test if µ is signicantly dierent from$21, which is the average bill for current AT&T customers.(a) The statistician wants to formulate hypotheses in favor of AT&T. State the hypotheses.(b) Calculate the t-statistic. Under what signicance level, among 1%, 5% and 10%, is the null hypothesis rejected?(c) Calculate the…

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