7. For a bivariate normal distribution with \( \mu_X, \mu_Y, \sigma_X^2 > 0, \sigma_Y^2 > 0 \) and \( |\rho| < 1 \),(a) Find the conditional density for \( X|Y = y \).(b) Determine the distribution of\[\frac{X^2 - 2\rho XY + Y^2}{1 - \rho^2}\]when \( \mu_X = \mu_Y = 0 \) and \( \sigma_X^2 = \sigma_Y^2 = 1 \).

Answered: Image /qna-images/answer/e838b6aa-a23c-480d-a6d2-3c999dc44537.jpg

Answered: 7. For a bivariate normal distribution…

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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The pressure of air (in MPa) entering a compressor is measured to be X = 8.5 ± 0.2, and the pressure of the air leaving the compressor is measured to be Y = 21.2 ± 0.3. The intermediate pressure is therefore measured to be P = √XY = 13.42 . Assume that X and Y come from normal populations and are unbiased. a) From what distributions is it appropriate to simulate values X* and Y*? b) Generate simulated samples of values X*, Y*, and P*. c) Use the simulated sample to estimate the standard deviation of P. d) Construct a normal probability plot for the values P*. Is it reasonable to assume that P is approximately normally distributed? e) If appropriate, use the normal curve to find a 95% confidence interval for the intermediate pressure.
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5. Assume you are shooting at a target centred at the origin (0,0), but bullets land at a random point with horizontal (X) and vertical (Y) coordinates following in- dependent standard Normal (0,1) distributions. Define the distance from the tar- get to be D = VX²+Y². For independent standard Normals, we already know that X? - x²(1) = Gamma(1/2,1/2) and X² + Y² ~ x²(2) = Gamma(1, 1/2). Y /D %3D (a) Find the probability that a bullet lands in the top right quadrant. (Hint: You don't need to calculate any integrals.) (b) Find the conditional probability that a bullet lands more than 1 unit away from the centre (D > 1), given Y = X, i.e. the bullet lands on the identity line; express the probability as a function of the standard Normal CDF ¤(·). (c) Find the CDF of X² + Y² in closed form. (d) Find the probability that a bullet lands more than 1 unit away from the centre, P(D> 1). (e) Find the CDF of D based on the CDF of X? + Y². (f) Use your answer in the previous part to find the PDF of…
Let the random variable X have'the marginal density 1 f, (x)= 1, 1
Write a function mysquares[v, m, μ, o] that constructs m samples of v sums-of-squares of the deviation from the mean (as in the workbook) with the X; drawn from the normal distribution N(μ, o). Also write histxsq[v, m, µ, σ] to plot a PDF histogram of your samples (with Automatic bspec), with the appropriate x² PDF plotted over the top. With m = 10 000, plot a few examples to see how well the x² distribution fits your samples. [Say v= 2, µ = 1, σ=2; v = 6, μ = 3, 0 = 10; v = 16, μ = 0, 0 = 1.]

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7. For a bivariate normal distribution with µx,µy,o} > 0,0² > 0 and |p| < 1, (a) Find the conditional density for X Y = y. (b) Determine the distribution of X² − 2pXY+Y² 1 - p² = 1. when μx = μy = 0 and o =