Solved Problems

Problem 1: Covariance Calculation (Basic)

Two variables XX (Temperature, ∘^{\circ}C) and YY (Expansion, mm) have the following joint distribution:
  • (20, 1): P=0.2P=0.2
  • (30, 2): P=0.5P=0.5
  • (40, 3): P=0.3P=0.3
Calculate the covariance between Temperature and Expansion.

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Problem 2: Marginal Distributions (Intermediate)

A civil engineering firm evaluates two suppliers of steel rebars based on two criteria: Grade XX (1 for Standard, 2 for High Strength) and Delivery Time YY (1 for On-Time, 2 for Delayed). The joint probability distribution f(x,y)f(x, y) is given as a table:
  • f(1,1)=0.40f(1,1) = 0.40
  • f(1,2)=0.15f(1,2) = 0.15
  • f(2,1)=0.30f(2,1) = 0.30
  • f(2,2)=0.15f(2,2) = 0.15
Find the marginal distributions of Grade XX and Delivery Time YY, and determine if the two variables are independent.

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Problem 3: Continuous Joint Density Function (Advanced)

The joint probability density function of two continuous variables XX (wind speed, m/s) and YY (structural vibration amplitude, mm) is defined as:
f(x,y)=k(x+y) for 0≤x≤2, and 0≤y≤2f(x,y) = k(x+y) \text{ for } 0 \le x \le 2, \text{ and } 0 \le y \le 2
Determine the value of the constant kk that makes this a valid joint PDF.

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