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Discrete independent random variables X and Y are given by the following laws of distribution:

X 0 3 Y - 2 - 1 2
P 0,3 0,7 P 0,2 0,4 0,4
Find M (X + Y) by two ways: 1) composing the law of distribution of X + Y; 2) using the property: M (X + Y) = M (X) + M (Y).

1 answer

To find the expected value \( M(X + Y) \) using both methods, we'll start by identifying the distributions of \( X \) and \( Y \).

Distribution of X:

  • \( X = 0 \) with \( P(X = 0) = 0.3 \)
  • \( X = 3 \) with \( P(X = 3) = 0.7 \)

Distribution of Y:

  • \( Y = -2 \) with \( P(Y = -2) = 0.2 \)
  • \( Y = -1 \) with \( P(Y = -1) = 0.4 \)
  • \( Y = 2 \) with \( P(Y = 2) = 0.4 \)

Step 1: Calculating \( M(X + Y) \) by composing the law of distribution of \( X + Y \)

First, we need to find the possible values of \( X + Y \) and their corresponding probabilities.

Possible combinations of values for \( X + Y \):

  1. When \( X = 0 \):

    • \( Y = -2 \) ⇒ \( X + Y = 0 - 2 = -2 \) with probability \( P(X = 0) P(Y = -2) = 0.3 \times 0.2 = 0.06 \)
    • \( Y = -1 \) ⇒ \( X + Y = 0 - 1 = -1 \) with probability \( P(X = 0) P(Y = -1) = 0.3 \times 0.4 = 0.12 \)
    • \( Y = 2 \) ⇒ \( X + Y = 0 + 2 = 2 \) with probability \( P(X = 0) P(Y = 2) = 0.3 \times 0.4 = 0.12 \)

  2. When \( X = 3 \):

    • \( Y = -2 \) ⇒ \( X + Y = 3 - 2 = 1 \) with probability \( P(X = 3) P(Y = -2) = 0.7 \times 0.2 = 0.14 \)
    • \( Y = -1 \) ⇒ \( X + Y = 3 - 1 = 2 \) with probability \( P(X = 3) P(Y = -1) = 0.7 \times 0.4 = 0.28 \)
    • \( Y = 2 \) ⇒ \( X + Y = 3 + 2 = 5 \) with probability \( P(X = 3) P(Y = 2) = 0.7 \times 0.4 = 0.28 \)

Summary of \( X + Y \) values and their probabilities:

  • \( X + Y = -2 \): Probability = \( 0.06 \)
  • \( X + Y = -1 \): Probability = \( 0.12 \)
  • \( X + Y = 1 \): Probability = \( 0.14 \)
  • \( X + Y = 2 \): Probability = \( 0.12 + 0.28 = 0.40 \) (combined from \( X=0, Y=-1 \) and \( X=3, Y=-1 \))
  • \( X + Y = 5 \): Probability = \( 0.28 \)

So the final probability distribution of \( X + Y \) is:

  • \( -2 \): \( 0.06 \)
  • \( -1 \): \( 0.12 \)
  • \( 1 \): \( 0.14 \)
  • \( 2 \): \( 0.40 \)
  • \( 5 \): \( 0.28 \)

Calculate \( M(X + Y) \):

\[ M(X + Y) = (-2)(0.06) + (-1)(0.12) + (1)(0.14) + (2)(0.40) + (5)(0.28) \] \[ = -0.12 - 0.12 + 0.14 + 0.80 + 1.4 = 2.1 \]

Step 2: Using the property \( M(X + Y) = M(X) + M(Y) \)

Calculate \( M(X) \):

\[ M(X) = (0)(0.3) + (3)(0.7) = 0 + 2.1 = 2.1 \]

Calculate \( M(Y) \):

\[ M(Y) = (-2)(0.2) + (-1)(0.4) + (2)(0.4) = -0.4 - 0.4 + 0.8 = 0 \]

Combine them:

\[ M(X + Y) = M(X) + M(Y) = 2.1 + 0 = 2.1 \]

Conclusion

Both methods yield the same expected value: \[ \boxed{2.1} \]