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Data Interview Question

Calculating the Expected Value in a Gaussian Sum

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Solution & Explanation

Problem Statement: We are given two Gaussian random variables, X1X_1 and X2X_2, with means m1m_1 and m2m_2, and variances v1v_1 and v2v_2 respectively. We know that the sum X1+X2=nX_1 + X_2 = n. We need to determine the expected value of X2X_2.

Understanding the Problem:

  • The sum X1+X2=nX_1 + X_2 = n implies a linear relationship between X1X_1 and X2X_2.
  • The expected value of a Gaussian random variable is its mean.

Key Concept: Linearity of Expectation

  • The linearity of expectation states that for any two random variables XX and YY, E[X+Y]=E[X]+E[Y]E[X + Y] = E[X] + E[Y]. This property holds regardless of whether the random variables are independent.

Solution:

  1. Setup the Equation:

    • We know X1+X2=nX_1 + X_2 = n.
    • By the linearity of expectation, we have: E[X1+X2]=E[X1]+E[X2]=nE[X_1 + X_2] = E[X_1] + E[X_2] = n
    • Substitute the known expected values (means) of X1X_1 and X2X_2: m1+E[X2]=nm_1 + E[X_2] = n
  2. Solve for E[X2]E[X_2]:

    • Rearranging the equation gives: E[X2]=nm1E[X_2] = n - m_1

Explanation:

  • The solution utilizes the linearity of expectation, which simplifies the calculation.
  • The fact that X1X_1 and X2X_2 are Gaussian is not directly relevant to finding the expected value of X2X_2 given X1+X2=nX_1 + X_2 = n. This is because the linearity of expectation applies to all random variables, not just Gaussian ones.
  • The dependency introduced by X1+X2=nX_1 + X_2 = n is handled by the rearrangement of the linear expectation equation.

Conclusion:

  • The expected value of X2X_2 given the sum X1+X2=nX_1 + X_2 = n is nm1n - m_1. This result is derived purely from the properties of the expectation operator and does not depend on the Gaussian nature of the variables.

Note:

  • This type of problem is useful to test understanding of the linearity of expectation and familiarity with Gaussian properties, although the Gaussian property is not used here.