2. Consider the following joint probability mass function, fx.y(x, y) of the discrete random variables X and Y.10.150.1020.200.300.200.052.1 P(X + Y > 8) = ?2.2 Give the marginal pmf of X, f, (x) in tabular form.X123p(x)2.3 Give the marginal pmf of Y, fy (y) in tabular form.Y5p(y)

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Answered: 2. Consider the following joint…

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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For a certain psychiatric clinic suppose that the random variable X represents the total time (in minutes) that a typical patient spends in this clinic during a typical visit (where this total time is the sum of the waiting time and the treatment time), and that the random variable Y represents the waiting time (in minutes) that a typical patient spends in the waiting room before starting treatment with a psychiatrist. Further, suppose that X and Y can be assumed to follow the bivariate density function fXY(x,y)=λ2e−λx, 0<y<x, where λ > 0 is a known parameter value. (a) Find the marginal density fX(x) for the total amount of time spent at the clinic. (b) Find the conditional density for waiting time, given the total time. (c) Find P (Y > 20 | X = x), the probability a patient waits more than 20 minutes if their total clinic visit is x minutes. (Hint: you will need to consider two cases, if x < 20 and if x ≥ 20.)
Let Y1 = 0.5, Y2 = 0.25, Y3 = 0.75, Y4 = 0.25 and Y5 = 1.25 be a random sample of width 5 selected from the population with the following probability density function. Which of the following is the estimation value obtained by the moment method for the unknown q parameter of this population?
Suppose that Y1,Y2,Y3 denote a random sample from an exponential distribution with density function f(y) = Consider the following four estimators of θ: ?1θe−y/θ, y>0,0, otherwise. θˆ =Y, θˆ =Y1+Y2, θˆ =Y1+2Y2, θˆ =Y1+Y2+Y3 =Y ̄. 11223343 Which estimators are unbiased? Among the unbiased estimators, which has the smallest variance?
Suppose that the random variables X,Y, and Z have the joint probability density function f(x,y,z) = 8xyz for 0<x<1, 0<y<1, and 0<z<1. Determine P(X<0.7).
Let X and Y be discrete random variables with joint pdf f(x,y) given by the following table: y = 1 y = 2 y = 3 x = 1 0.1 0.2 0 x = 2 0 0.167 0.4 x = 3 0.067 0.022 0.033 Find the marginal pdf’s of X and Y. Are X and Y independent?
If we let RX(t) = ln MX(t), show that R X(0) = μ and RX(0) = σ2. Also, use these results to find the mean and the variance of a random variable X having the moment-generating function MX(t) = e4(et−1)
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…
Suppose that Y1, . . . , Yn is a random sample from a population whose density function is
2. Identify the probability density function, then find the mean and variance without integrating. b. f(x) =1/6 e^−x/6, [0,∞) c. f(x) =1 / 3√2π e^−(x−16)^2/18, (−∞,∞)
Suppose X and Y are independent and identically distributed (i.i.d.) randomvariables, each with the uniform distribution on [0, 1]. What is the cumulative distributionfunction and the density function of XY ?
If the independent random variables X and Y havethe marginal densitiesf(x) =⎧⎪⎪⎨⎪⎪⎩12for 0 < x < 20 elsewhereπ(y) =⎧⎪⎪⎨⎪⎪⎩13for 0 < y < 30 elsewherefind(a) the joint probability density of X and Y;(b) the value of P(X2 + Y2 > 1).
Let X denote the reaction time, in seconds, to a certain stimulus and Y denote the temperature (◦F) at which a certain reaction starts to take place. Suppose that two random variables X and Y have the joint density

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