If two random variables, X and Y, are independent, then the joint characteristic function is equal to the product of their individual characteristic functions. O» (@,0,) = 0x(@,) 0, (@,)

Q: The marginal distribution function of X and Y are Fx(x) = [[Fxy(x, y) dx dy X,Y and Fy(y) = [[_Fx₁y…

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Q: f A is an event in the finite sample space, then prove that P (A) equals the sum of the…

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Q: Example 28: If u is moment about origin, prove the rth that r-1 Σ H-j Kj; where Kj is the jth…

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Q: If 5|(2a + 1) for some integer a, then 25|(14a² + 19a – 19).

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Q: Prove the following: Given: AC1 BD,D is the midpoint of AC Prove: AABD = ACBD A

A: Provides figure is  AC¯ ⊥ BD¯ D is the midpoint of AC¯.   To prove: ∆ABD≅∆CBD

Q: Autocorrelation function is maximum at the origin, i.e., |Rxx (7) |S Rxx (0) XX

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Q: If X and Y are two random variables, then the covariance of x+a,Y +b,where a and b are constant is…

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Q: The properties of the cross power spectrum for real random processes X(t) and Y(t) are given by (1)…

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Q: If X(t) and Y(t) are two random processes with autocorrelation functions Rx (7) and Ryy (7), then…

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Q: The probability of the impossible event is zero, i.e., if ø is the subset (event) containing no…

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Q: Difference of two independent Poisson processes is not a Poisson process

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Q: If Sxx (@) is a psd of the WSS random process X(t), then XX 1 277 [ Sxx (@)dw = A{E[X²(1)]} = R₂x…

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Q: 10. The square of any integers is of the form 5k, 5k + 1 or 5k + 4.

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Q: Prove that if z is on the annulus, ½ ≤ | z |≤ 1, then |Im (z−¹+z+0.25)| <ñ.

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Q: Fxy(x, y) is a monotonic and non-decreasing function of both x and y.

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Q: The marginal moment generating function can be obtained from the joint moment generating function as…

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Q: If øx(@) is a characteristic function of a random variable X, then , (co) = 6, (@), where c is a…

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Q: (8) If a random process X(t) with zero mean has the DC component A as Y(t) = A+X(t), then Ryy (T) =…

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Q: (5) If X is any random variable, then the inequality, \E[X]<E\X\ exists

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Q: (5) If X(t) and Y(t) are uncorrelated and have constant mean values X and Y, then Sxy (@) - 2πX δω

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Q: Example 4: If x1, X2 , .… , Xn are random observations on a Bernoulli variable X ...., aking the…

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Q: (1) The mean square value of X(t) is E[X?(t)] = Rxx(0) It is equal to the power of the process X(t)…

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Q: The probability of the joint event {x, < X ≤ x₂,₁ <Y ≤y₂} is given by P{x <X<x,,y<Y<y,} = Fxy(x,…

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Q: IV. The inter-arrival time of a Poisson process with parameter ↑ has an exponential distribution…

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Q: The joint distribution function is equal to Fx,y (x, y) = f* fx₁y (x, y) dx dy

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Q: (ii) An(BUC) = (AN B) U (A nC). i.e., union and intersection are distributive over intersection and…

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Q: (1) The characteristic function is unity at w = 0, and given by Øx (@o = ¢x(0) = 1 la=0

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Q: The difference of two independent Poisson variates is not a Poisson vairate.

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Q: The power density spectrum of a real process X(t) is an even function i.e., Sxx (-a)=Sxx (@) X(t) is…

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Q: Example 6-26. If the possible values of a variate X are 0, 1, 2, 3, ... then 00 E(X) = 2 P(X> n) n=0

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Q: (1) Rxy(t) = Ryx(-t) is a symmetry property.

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Q: The imaginary part of Sxy (@) and the imaginary part of Sy (@) are odd functions of w, i.e., Im[Sxy…

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Q: Example 3: If x1, X2 , ... , Xn is a random sample from a normal population .2 N(µ,1), show that t=-…

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Q: 2.) The 'r'th cumulant of the sum of independent random variables is equal to the sum of the 'r'th…

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Q: V. If the number of occurrences of an event E in an interval of length t is a Poisson process X (t)…

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