Let X denotes the reaction time, in seconds; to a certain stimulants and Y denote the temperature at which a certain reaction starts to take place. Suppose that two random variables X and Y have the joint density J 4xy, f (x, y) = 0, 0
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- 3. Let X and Y be random variables with joint density function. fxy(x, y) = a(x+2y), 0 5V). 2vQ 4.2. Let (X, Y) be a random variable with the following density: 15x²y 0 0 < x < y < 1, otherwise. fx,y(x, y) = { { 1. Express E(YX) in terms of X. 2. Express Var(X|Y) in terms of Y.A random process {X(t)} is given by X(t) = A cos pt + B sin pt, where A and B are independent RVs such that E(A) = E(B) = 0 and E(A²) = E(B²) = o². Find the power spectral density of the process I %3DProb ... solve Q 5 & Q 6Let Y1, Y2, ..., Yn denote a random sample of size n from a population whose density is given by 3y2 0 < y< 0, f (y|0) = өз 0, elsewhere. 1.1. Show that Y(n) max(Y1, Y2, ... , Yn) is sufficient for 0.Q 4.2. Let (X, Y) be a random variable with the following density: {152-2 0 £x,x (x, y) = { 1. Express E(Y|X) in terms of X. 2. Express Var(XY) in terms of Y. 0 < x < y < 1, otherwise.Here is another infectious disease model. Once a person becomes infected, the time X, in days, until the person becomes infectious (can pass on the disease) can be modeled as a Weibull random variable with density function f(x,α,β) = (α/βα)xα−1e−(x/β)α for 0 ≤ x ≤ ∞ and 0 otherwise with α = 3.7 and β = 7.1α is the shape parameter and β is the scale parameter. Hint: Solve this with the built-in R functions for the Weibull distribution (dweibull(),pweibull(), qweibull()) not f as defined above. Otherwise you may get intermediate values too large to use. For a) and b) the text (and notes) give formulas for the answers. You can calculate from these formulas. Note that these formulas use the gamma function. Γ(α) is the gamma function. In R, there is a built-in function gamma() which calculates this.d) What is the probability that X is larger than its expected value?e) What is the probability that X is > 2?Consider a particle to be constrained to lie along a one-dimensional segment 0 to a. The probability that the particle is found to lie between x and x+dx is given by p(x)dx= = 2 - a sin² (n) da, (1) where n 1, 2, 3, . . . . = .. 1) Show that p(x) is normalized. 2) Calculate the average position of the particle along the line segment. 3) Calculate the variance, σ², associated with p(x).Define two random processes by X (1) = A cos (@nt+ 0) Y (t) = Z (t) · cos (@, t + 0), where A, on are real positive constants, and 0 is a random variable independent of Z (t) which is a random process with a constant mean Z. Show that the cross spectral density. A Zn 2 regardless of the form of the Pdf of 0. Sxy (m) = (Co +0)g + (m - 0)g]tEx. 4. Let X,Y are independent random variables where X ~ N(0, 1) and Y ~ G(1/2, n/2). Let T:= X//Y/n, then show that the density function of T is described as r(n+1)/2) (1+) VnnI'(n/2) -(n+1)/2 fr(t) - (-∞The joint density function of two continuous random variables X and Y is expressed Q3 below. Find: (a) the value of the constant c, (b) P(X 2 3,Y < 4), (c) for the two possible solutions, only sketch the integration boundaries and write the integration equation for P(Y + 2X < 9). cos(x) y- 2 1sxS4,3 syS5 f(x,y) = otherwiseE The density function of a continuous Random variable X is fx (x) = ax 0 Sx <1 for for 1SEE MORE QUESTIONSRecommended textbooks for youA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSONA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON