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Trigonometry (MindTap Course List)

Trigonometry (MindTap Course List)

10th Edition

ISBN:9781337278461

Author:Ron Larson

Publisher:Ron Larson

Chapter6: Topics In Analytic Geometry

Section6.4: Hyperbolas

Problem 5ECP: Repeat Example 5 when microphone A receives the sound 4 seconds before microphone B.

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12 A random process has sanmple functions of the form x(t) = A cos (o t+ 0) in which
A and o are constants and 0 is a random variable.
(i) Prove that this process is stationary in the wide sense if 0 is uniformly distributed
between 0 and 2n.

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Repeat Example 5 when microphone A receives the sound 4 seconds before microphone B.
3 Show that the random process X(t) = A cos (@n t + 0) is wide-sense stationary if it is assumed that A and on are constants and 0 is a uniformly distributed random variable on the interval (O, 2n).
2 Let X (t) be a random process with mean 3 and auto correlation R(t, t2) = 9 + 4 e-0.2 ,-t, Determine the mean, variance and covariance of the random variables Z =X (5) and wX (8).
Let X and Y be independent random variables with common moment generating function 2 1 M(t) 1-2t Compute E[(X + Y)²].
Q. 3 Let continuous random variables X and Y be independent identically distributed random variables with the following respective pdfs: fx (x) = 2e-2x; x > 0 and fy (y) = 2e-2y; y > 0 Define a new random variable Z = X + Y. Derive moment %3D generating function of Z, i.e. Mz(t).
(ii) Suppose that X₁ and X₂ have joint pdf 2, 10, Compute the joint pdf of random variables Y₁ = X₁ and Y₂ = X2. f(x1, x₂): for 0 < x1 < x₂ < 1 otherwise =
16) A stationary random process has an autocorrelation function of Rx (T)=16-e cos 20rt+8 cos10TT. Find the variance of this process.
B) Let the random variable X have the moment generating function e3t M(t) for -1
Let X be a discrete random variable with pmf given by f(x = k) = k = 2,3,4,... ak = a(a – 1) Find the moment generating function of X Select one: а. a - 1 a -t b. a a -t C. (a – 1) e2t - a - t O d. (a – 1)e2"
(6) Let X be a Poisson random variable with parameter 0, then E(X²) = 1. От T O F
Let the joint pdf of random variables X,Y be fx,y (x, y) = a(x + y)e-2-Y, for all æ > 0, y 2 0. Find the conditional pdf Tylx (y|x) = &x2) fx(x)
A random process is described by X(t) = A , where A is a continuous random variable uni formly distributed on (0,1). Show that X(1) is stationary process.

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