In the transmission of digital information, the probability that a bit has high, moderate,
and low distortion is 0.01, 0.04, and 0.95, respectively. Suppose that three bits are transmitted and that
the amount of distortion of each bit is assumed to be independent. Let X and Y denote the number of
bits with high and moderate distortion out of the three, respectively
(a) fX,Y (x, y)
(b) fX(x)
(c) E[X]
(d) fY |X=3(y)
(e) E[Y | X = 3]
(f) Var(Y | X = 3)
(g) Are X and Y independent?
3e
To solve this problem, we can use the concept of probability mass function (PMF), expected value, conditional probability, and independence. Let's break down each part of the question:
(a) fX,Y(x, y):
To find the joint PMF, we need to calculate the probability of each combination of X (number of bits with high distortion) and Y (number of bits with moderate distortion) occurring. Since X and Y are assumed to be independent, we can use the multiplication rule:
fX,Y(x, y) = P(X = x and Y = y) = P(X = x) * P(Y = y)
Given the probabilities provided:
P(X = 0) = 0.01^3
P(Y = 0) = 0.95^3
P(X = 1) = 3 * 0.01^2 * 0.99
P(Y = 1) = 3 * 0.95^2 * 0.04
P(X = 2) = 3 * 0.01 * 0.99^2
P(Y = 2) = 3 * 0.95 * 0.04^2
P(X = 3) = 0.99^3
P(Y = 3) = 0.04^3
Now we can calculate fX,Y(x, y) for each combination of x and y.
(b) fX(x):
To find the marginal PMF of X, we need to sum up the probabilities over all possible values of Y.
fX(x) = Σ fX,Y(x, y) for all y
(c) E[X]:
The expected value of X is the sum of the product of each value of X with its corresponding probability.
E[X] = Σ x * fX(x) for all x
(d) fY|X=3(y):
To find the conditional PMF of Y given X = 3, we divide the joint PMF by the probability of X = 3.
fY|X=3(y) = P(Y = y | X = 3) = P(X = 3 and Y = y) / P(X = 3)
(e) E[Y|X=3]:
The expected value of Y given X = 3 can be calculated using the conditional PMF.
E[Y|X=3] = Σ y * fY|X=3(y) for all y
(f) Var(Y|X=3):
To find the variance of Y given X = 3, we need to calculate the conditional variance.
Var(Y|X=3) = Σ (y - E[Y|X=3])^2 * fY|X=3(y) for all y
(g) Are X and Y independent?
X and Y are independent if fX,Y(x, y) = fX(x) * fY(y) for all x and y. Check if this condition holds true.
By following these steps, you should be able to solve the given problem.