A random process X(t) is determined by tossing two dice to determine which of the following sinusoids to pick
X(t) = x1(t) = sin(t), if sum is 3 or 7.
= x2(t) = cos(2t), if the sum is 2 or 12.
= x3(t) = sin(2t), otherwise.
Find E(X) and variance for X= π/2.
Answers
Given : A random process X(t) is determined by tossing two dice to determine which of the following sinusoids to pick
X(t) = x1(t) = sin(t), if sum is 3 or 7.
= x2(t) = cos(2t), if the sum is 2 or 12.
= x3(t) = sin(2t), otherwise.
To Find : E(X) and variance for X= π/2.
Solution:
Sum 3 or 7 (1 , 2) , (2 , 1) , ( 1 , 6) , ( 2 , 5) , (3, 4) , ( 4, 3) , ( 5 , 2) , ( 6 , 1)
Probability = 8/36 = 2/9
Sum 2 or 12 ( 1, 1 ) ( 6 , 6) Probability = 2/36 = 1/18
Otherwise = 1 - 1/18 - 2/9 = 13/18
sin(t) => sin(π/2) = 1 Probability = 2/9
cos(2t) => cos (2(π/2)) = - 1 Probability = 1/18
sin(2t) => cos (2(π/2)) = 0 Probability = 13/18
X P(X) X.P(X) X² X²P(X)
1 2/9 2/9 1 2/9
-1 1/18 -1/18 1 1/18
0 13/18 0 0 0
E(X) = ∑ X.P(X) = 2/9 - 1/18 = 3/18 = 1/6
E(X) = 1/6
Variance = ∑X²P(X) - ( ∑ X.P(X))²
= 5/18 - (1/6)²
= 5/18 - 1/36
= 9/36
= 1/4
Learn More"
A r.v. X has the given probability distribution, Find the value of k and ...
https://brainly.in/question/6536531