
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
As a result of processing data from many years of observations, distributions of the random variables $X$ and $Y$ were obtained. These represent the number of households in each of two districts in a region where the yield of spring cereals may exceed 35 quintals per hectare.

For the first district:

$$
X: \begin{array}{cccc}
x_{i} & 1 & 2 & 3 \\
p\left(x_{i}\right) & 0,1 & 0,6 & 0,3
\end{array}
$$

For the second district:

$$
Y: \begin{array}{ccc}
y_{j} & 0 & 1 \\
p\left(y_{j}\right) & 0,2 & 0,8
\end{array}
$$

Find the expected value $M(Z)$ and the variance $D(Z)$ of the random variable $Z = X + Y$ using two methods:

(a) Based on the distribution law of $Z$;

(b) Using the properties of the expected value and variance, as reflected in formulas (33) and (42).

Verify that the properties of numerical characteristics for independent random variables are fulfilled under the conditions of this problem.

Present the answer in LaTex format: \boxed{Your answer}
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