Let U,V be independent Uniform [0, 1] random variables. Find the CDF and PDF of each of the following random variables: (a) U+V (b) |U – V| (c) min{U,V} (d) max{U,V}.
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Let U,V be independent Uniform [0,1] random variables. Find the CDF and
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- Let X and Y be two random variables, denoting the age and weight (in kg), respectively. Consider a random sample of size n = 20 from these two variables: X = {28, 50, 61, 72, 80, 51, 20, 35, 28, 97, 37, 64, 46, 67, 34, 21, 21, 59, 46, 46} Y = {62, 87, 88, 109, 95, 104, 58, 82, 70, 102, 67, 104, 83, 89, 69, 68, 65, 103, 78, 102} (a) Find the mean and median of X. (b) Find the variance of X. (Use population variance, i.e normalize by total number of observations) (c) Find the 2-dimensional mean μ (d) Find the covariance matrix between these two variables. (Do not use the built-in function) (e) Find the eigenvalues and eigenvectors of the covariance matrix. (Do not use the built-in function) (f) What is the probability of observing an age of 75 or higher? (g) What is the correlation between age and weight? (Do not use the built-in function) (h) Draw the scatter plot of the variable Y versus the variable X.Let ã be a random variable with an invertible cdf F. What is the distribution of Fa(ã)? Justify your answer mathematically.For two random variables X andY the following information is given. V(X)=3, V(Y)=2 and cov(X, Y)= -1. Calculate V(2X+Y). Select one: a. 5 b.16 С. 12 d. 10
- Match up the following random variables with the values they can take. X~ Bin(5, 0.7) Drag answer here X 0,1, 2, ... X~ Geom(0.7) X = 0,1, 2, 3, 4, 5 Drag answer here X~ Bernoulli(0.7) X = 0, 1 Drag answer here X~Bin(1,0.7) X = 1, 2, 3, ... Drag answer here X~Poisson(5) Drag answer hereProve a formula for the moment-generating function (mgf) of Y in terms of the mgf's of X₁ and N. For any random variable Z, the mgf is defined as oz(s) = E[e³Z], where s € IR and whenever the expectation is defined.Exercise 14 pls
- Let X1, X2, X3, X4 be independent random variables of size 4 from a gamma distribution G(a, X) = ra) -xa-le-a with a = 6 and Find the expected value of Y = X; for i = 1, 2, 3, 4. %3D Answer:2. Some properties of Expected value and variance of a random variable. a) Assume that X is an arbitrary discrete random variable, and a and b are constant. Using the definitic Show: and E(aX + b) = a · E(X) + b V(aX + b) = a² · V(X ) Stat 3128 Ali Mahzarnia P STAT 3128 Ali Mahzarnia b) Justify the computational formula of Variance of a random variable which is to justify : V(X) = E[(X – µ°] = Ex – µ)P • ptx) = | 2: Here needs justification By Cauchy Schwarz inequality it can be shown that the right hand side is always positive. Analogs expression in Mechanic : Parallel axis theorem Iem = I– md² Moment of Moment of inetria about Inertia of an an axis shifted object about the center of a mass by d from center of mass (a parallel shift) Icm is dispersion around the mean and is like second central moment (variance) I is like second moment if d is mean m is like sum total all the weight of each of the x which all add up to 1 d squared is like squared of mean since we,prove
- he joint pat of the random variable X and Yis iven in the below matrix. Find the mean value f X and Y: P(X,Y)= X 2.45 and 2.1 1.9 and 2.45 O 1.9 and 2.1 2.1 and 2.45 Y 0.05 0.05 0.1 0.05 0.1 0.35 0 0.2 0.1A toll road charges $ 4 for passenger cars and $ 7 for othervehicles. LetX1be the number of passenger cars and X2 be the number ofother vehicles entering the toll road in one hour. Assume thatX1andX2are independent normal random variables with means μ1= 45, μ2= 20, andstandard deviations σ1= 5, σ2= 6. a) Express the toll road revenue per hourY as a function of X1 and X2. b) Find the mean μY and the standard deviation σY of Y. c) Find the probability that the toll road revenue during a particular hour is between $300 and $400 inclusive.X, Y and Z are independent geometric random variables with parameters p1, p2 and p3.a. Find the expected value of the minimum of the three random variables.b. Find the expected value of the second smallest of the three random variables