Question

Problem II (50pts) (From a pair of Random Variables to a pair of Random Signals) Let us define a pair of random signals, X(t) and Y(t), as sinusoidal signals with random amplitude R and random phase ?, as follows, X(t) = R · cos(2?f?t + ?), Y(t) = R · sin(2?f?t + ?). Here, the parameter f? (cycles/second, or Hz) is a fixed deterministic constant, termed as carrier frequency in wireless communications applications. Note that randomness in the pair of random signals {X(t), Y(t)} is induced by a pair of random variables {R, ?} with the following joint probability density function (pdf), p_{R,?}(r, ?) = rac{r}{2??²} · exp(-rac{r²}{2?²}), 0 ? ? ? 2?, 0 ? r < ?. ?² is simply a parameter. (1). (10pts) Find the marginal pdfs of random variables R and ?, respectively. That is, p_R(r) = ? p_?(?) = ? (2). (10pts) Find the conditional pdfs of random variables R and ?, respectively. That is, p_{R|?=?}(r|?) = ? p_{?|R=r}(?|r) = ? (3). (20pts) Find the following statistics of random signals X(t) and Y(t), respectively, (2pts)The mean signal: E{X(t)} = ? (2pts) The mean signal: E{Y(t)} = ? (2pts) The variance of X(t): Var{X(t)} = ? (2pts) The variance of Y(t): Var{Y(t)} = ? (4pts) The ACF of X(t): E{X(t) · X(t - ?)} = ? (4pts) The ACF of Y(t): E{Y(t) · Y(t - ?)} = ? (4pts) The CCF btw X(t) and Y(t): E{X(t) · Y(t - ?)} = ? (4). (10pts) Using the above results to find the probabilities P(A), P(B), P(A · B), P(A + B) and P(A|B), where the events or subsets A and B are defined as follows, A ? {{X(t), Y(t)}: X²(t) + Y²(t) ? ?²}, B ? {{X(t), Y(t)}: ?² ? X²(t) + Y²(t) ? 2?²}.

          Problem II (50pts) (From a pair of Random Variables to a pair of Random Signals)

Let us define a pair of random signals, X(t) and Y(t), as sinusoidal signals with random
amplitude R and random phase ?, as follows,

X(t) = R · cos(2?f?t + ?), Y(t) = R · sin(2?f?t + ?).

Here, the parameter f? (cycles/second, or Hz) is a fixed deterministic constant, termed as
carrier frequency in wireless communications applications.
Note that randomness in the pair of random signals {X(t), Y(t)} is induced by a pair of
random variables {R, ?} with the following joint probability density function (pdf),

p_{R,?}(r, ?) = rac{r}{2??²} · exp(-rac{r²}{2?²}), 0 ? ? ? 2?, 0 ? r < ?. ?² is simply a parameter.

(1). (10pts) Find the marginal pdfs of random variables R and ?, respectively. That is,
p_R(r) = ?
p_?(?) = ?

(2). (10pts) Find the conditional pdfs of random variables R and ?, respectively. That is,
p_{R|?=?}(r|?) = ?
p_{?|R=r}(?|r) = ?

(3). (20pts) Find the following statistics of random signals X(t) and Y(t), respectively,
(2pts)The mean signal: E{X(t)} = ?
(2pts) The mean signal: E{Y(t)} = ?
(2pts) The variance of X(t): Var{X(t)} = ?
(2pts) The variance of Y(t): Var{Y(t)} = ?
(4pts) The ACF of X(t): E{X(t) · X(t - ?)} = ?

(4pts) The ACF of Y(t): E{Y(t) · Y(t - ?)} = ?

(4pts) The CCF btw X(t) and Y(t): E{X(t) · Y(t - ?)} = ?

(4). (10pts) Using the above results to find the probabilities P(A), P(B), P(A · B),
P(A + B) and P(A|B), where the events or subsets A and B are defined as follows,
A ? {{X(t), Y(t)}: X²(t) + Y²(t) ? ?²}, B ? {{X(t), Y(t)}: ?² ? X²(t) + Y²(t) ? 2?²}.

Problem II (50pts) (From a pair of Random Variables to a pair of Random Signals)

Let us define a pair of random signals, X(t) and Y(t), as sinusoidal signals with random
amplitude R and random phase ?, as follows,

X(t) = R · cos(2?f?t + ?), Y(t) = R · sin(2?f?t + ?).

Here, the parameter f? (cycles/second, or Hz) is a fixed deterministic constant, termed as
carrier frequency in wireless communications applications.
Note that randomness in the pair of random signals X(t), Y(t) is induced by a pair of
random variables R, ? with the following joint probability density function (pdf),

pR,?(r, ?) = racr2??² · exp(-racr²2?²), 0 ? ? ? 2?, 0 ? r < ?. ?² is simply a parameter.

(1). (10pts) Find the marginal pdfs of random variables R and ?, respectively. That is,
pR(r) = ?
p?(?) = ?

(2). (10pts) Find the conditional pdfs of random variables R and ?, respectively. That is,
pR|?=?(r|?) = ?
p?|R=r(?|r) = ?

(3). (20pts) Find the following statistics of random signals X(t) and Y(t), respectively,
(2pts)The mean signal: EX(t) = ?
(2pts) The mean signal: EY(t) = ?
(2pts) The variance of X(t): VarX(t) = ?
(2pts) The variance of Y(t): VarY(t) = ?
(4pts) The ACF of X(t): EX(t) · X(t - ?) = ?

(4pts) The ACF of Y(t): EY(t) · Y(t - ?) = ?

(4pts) The CCF btw X(t) and Y(t): EX(t) · Y(t - ?) = ?

(4). (10pts) Using the above results to find the probabilities P(A), P(B), P(A · B),
P(A + B) and P(A|B), where the events or subsets A and B are defined as follows,
A ? X(t), Y(t): X²(t) + Y²(t) ? ?², B ? X(t), Y(t): ?² ? X²(t) + Y²(t) ? 2?².

Added by Magdalena P.

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