Question

Let Ω={0,1,2,3}, P({k})=1/4 for k=0,1,2,3. Define two random variables:

X(ω) = sin((2πω)/2)
Y(ω) = cos((2πω)/2)
Find the probability mass functions (PMFs) of the random variables X and Y.

Compute P({ω∈Ω:X(ω)=Y(ω)}).

Answer 1

The probability mass functions (PMFs) of the random variables X and Y are calculated by finding the probability of each possible outcome. The PMF of X is {0: 1/4, 1: 1/4, 2: 1/4, 3: 1/4} and the PMF of Y is {1: 1/4, 0: 1/4, -1: 1/4, 0: 1/4}. The probability P({omega in Omega: X(omega) = Y(omega)}) is 1/2.

Step-by-step explanation:

To find the probability mass functions (PMFs) of the random variables X and Y, we need to calculate the probability of each possible outcome. For X, we have four outcomes (0, 1, 2, 3), and for Y, we also have four outcomes (1, 0, -1, 0) corresponding to the sine and cosine values of the given omega values. The probability of each outcome is 1/4, so the PMF of X is {0: 1/4, 1: 1/4, 2: 1/4, 3: 1/4}, and the PMF of Y is {1: 1/4, 0: 1/4, -1: 1/4, 0: 1/4}.

To compute P({omega in Omega: X(omega) = Y(omega)}), we need to find the probability that the value of X is equal to the value of Y. Looking at the PMFs of X and Y, we can see that X(omega) = Y(omega) if and only if omega = 0 or omega = 3. Therefore, the probability is P({omega in Omega: X(omega) = Y(omega)}) = P({0, 3}) = P({0}) + P({3}) = 1/4 + 1/4 = 1/2.