Question

In the transmission of digital information, the probability that a bit has high, moderate, and low distortion is 0.01,0.03, and 0.96 , 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. Determine the following:

fY∣1​(y) Round your answers to four decimal places (e.g. 98.7654).

Answer 1

To determine the conditional probability mass function fY|1​​(y), we use the binomial distribution given one high distortion bit (X=1), leading to probabilities for Y=0, 1, and 2 as 0.9409, 0.0582, and 0.0009, respectively.

Step-by-step explanation:

To find the conditional probability mass function fY|1​​(y), given that there is one bit with high distortion (X=1), we will use the binomial distribution for the number of bits with moderate distortion (Y), given the probabilities. Since there are 3 bits in total and one bit is already with high distortion, we only have 2 trials left for moderate distortion.

Let's calculate the probability for each possible value of Y given X=1:

• For Y=0, which means no moderate distortion in the remaining two bits, the probability is (1-0.03)^2 = 0.9409.
• For Y=1, which means one of the two bits has moderate distortion, the probability is 2 * (0.03) * (1-0.03) = 0.0582.
• For Y=2, which means both remaining bits have moderate distortion, the probability is 0.03^2 = 0.0009.

These probabilities sum up to 1 as expected.