Calculating the distribution of the random variable Y at the outputThe random variable X at...

April Bush

April Bush

Answered

2022-06-27

Calculating the distribution of the random variable Y at the output
The random variable X at the entrance to the transmission path is sufficient for the following distribution.
X = P ( x )
x 1 = 0.1 x 2 = 0.2 x 3 = 0.3 x 4 = 0.4
The random variable Y at the output of the transmission link only takes three values. The transfer matrix T with p i j = P ( Y = y j | X = x i ) is
T = ( p i j ) = ( 0.50 0 0.50 0.20 0.40 0.40 0.50 0.25 0.25 0 0.50 0.50 )

Answer & Explanation

last99erib

last99erib

Expert

2022-06-28Added 19 answers

Step 1
What you have is a matrix of conditional probabilities, with the entry in row i, column j of T corresponding to P ( Y = y j | X = x i ). To find the marginal distribution of Y, use the law of total probability:
P ( Y = y j ) = i = 1 4 P ( Y = y j | X = x i ) P ( X = x i )
Step 2
Note: If p represents a 1 × 4 row vector summarizing the probability masses of X, then pT is a 1 × 3 row vector summarizing the masses of Y.

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