Question

A random process X(t) is defined based on a fair coin toss experiment as follows: X(t) = {cos(πt) if heads t if tails (a) Find E[X(t)) (b) Find the CDF Fₓₜ (x) for t = 0,1 = 1/2, t = 1.

Answer 1

The expected value of X(t) is calculated as the average of the two possible outcomes, cos(πt) and t, each with a probability of 0.5. The CDF of X(t) is a step function that changes based on the value of t, creating jumps at specific values derived from the outcomes of the coin toss.

Step-by-step explanation:

For a random process X(t) based on a fair coin toss, where X(t) = cos(πt) if heads and X(t) = t if tails, we calculate the expected value and cumulative distribution function (CDF) at different time points.

(a) Expected Value E[X(t)]

Since the coin is fair, we have P(heads) = P(tails) = 0.5. Thus, the expected value is:

E[X(t)] = 0.5 x E[cos(πt) | heads] + 0.5 x E[t | tails] = 0.5 x cos(πt) + 0.5 x t

(b) CDF FXt(x), for t = 0, t = 1/2, and t = 1

For t = 0, X(0) = cos(0) if heads and X(0) = 0 if tails. Since cos(0) = 1, CDF FX0(x) is a step function that jumps from 0 to 0.5 at x = 0 and to 1 at x = 1.

For t = 1/2, X(1/2) = cos(π/2) if heads and X(1/2) = 1/2 if tails. Since cos(π/2) = 0, CDF FX1/2(x) is a step function that jumps from 0 to 0.5 at x = 0 and to 1 at x = 1/2.

For t = 1, X(1) = cos(π) if heads and X(1) = 1 if tails. Since cos(π) = -1, FX1(x) is a step function that jumps from 0 to 0.5 at x = -1 and to 1 at x = 1.

Answer 2

The expected value E[X(t)] of the random process X(t) is calculated as 0.5 * cos(πt) + 0.5 * t. The Cumulative Distribution Function (CDF) at t = 0 is 0.5 for x >= 1 and at t = 1 is 0.5 for x >= 1, while the CDF is not defined for t = 1/2.

Step-by-step explanation:

The random process X(t) can result in either cos(πt) if a fair coin toss results in heads or t if it results in tails, each with probability 0.5.

Expected Value E[X(t)]

To find the expected value E[X(t)], we use the definition of expected value for discrete distributions:

E[X(t)] = Σpixi

Here, pi is the probability of the ith outcome, and xi is the value of X(t) for that outcome. For X(t), this means:

E[X(t)] = 0.5 * cos(πt) + 0.5 * t

Cumulative Distribution Function (CDF)

The CDF for a random variable at a given time t is the probability that the random variable is less than or equal to x. For the specified times t = 0,1, the CDF is:

For t = 0: for x >= cos(0) and x >= 0, since cos(0) = 1 and the other outcome t = 0.

For t = 1/2: Since cos(π(1/2)) = 0, the CDF at t = 1/2 is not well-defined for this process.

For t = 1: for x >= cos(π) and x >= 1, which simplifies to 0.5 for x >= 1 since cos(π) = -1.