Calculating the distribution of the random variable Y at the output

Question

Given

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

$T = (p_{i}^{j})= \begin{pmatrix} 0.50 & 0 & 0.50 \\ 0.20 & 0.40 & 0.40 \\ 0.50 & 0.25 & 0.25 \\ 0 & 0.50 & 0.50 \end{pmatrix}$

Task

Calculate the distribution of the random variable Y at the output.

Approach

I know that $P(y_j | 0.1) = 0.50$ $P(y_j | 0.2) = 0.20$... for the first column but I don't know how to solve the equation that the condition is fulfilled.

Answer 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:

$$P(Y=y_j)=\sum_{i=1}^4 P(Y=y_j|X=x_i)P(X=x_i)$$

Note: If $p$ represents a $1\times 4$ row vector summarizing the probability masses of $X$, then $pT$ is a $1\times 3$ row vector summarizing the masses of $Y$.