Solved Problems

Two variables $X$ (Temperature, $^{\circ}$C) and $Y$ (Expansion, mm) have the following joint distribution:

• (20, 1): $P=0.2$
• (30, 2): $P=0.5$
• (40, 3): $P=0.3$

Calculate the covariance between Temperature and Expansion.

A civil engineering firm evaluates two suppliers of steel rebars based on two criteria: Grade $X$ (1 for Standard, 2 for High Strength) and Delivery Time $Y$ (1 for On-Time, 2 for Delayed). The joint probability distribution $f(x, y)$ is given as a table:

• $f(1,1) = 0.40$
• $f(1,2) = 0.15$
• $f(2,1) = 0.30$
• $f(2,2) = 0.15$

Find the marginal distributions of Grade $X$ and Delivery Time $Y$, and determine if the two variables are independent.

Using the joint probability distribution from Problem 2, find the conditional probability that the delivery is delayed ($Y=2$) given that the steel grade is High Strength ($X=2$).

The cost of materials for a project depends on the quantity of cement ($X$, in tons) and the quantity of sand ($Y$, in tons). The expected quantities are $E[X] = 50$ tons and $E[Y] = 120$ tons. The cost function is $C = 100X + 20Y$. Calculate the expected total cost, $E[C]$.

Using the data from Problem 1, calculate the correlation coefficient between Temperature ($X$) and Expansion ($Y$). The standard deviations are given as $\sigma_X = 7.746$ and $\sigma_Y = 0.7000$.

The joint probability density function of two continuous variables $X$ (wind speed, m/s) and $Y$ (structural vibration amplitude, mm) is defined as:

Determine the value of the constant $k$ that makes this a valid joint PDF.

Given the joint PDF from Problem 6, where $k = 1/8$:

Determine the marginal probability density function of wind speed, $f_X(x)$.

Let $X$ and $Y$ have the joint PDF from Problem 7:

Find the expected value of the product $XY$, $E[XY]$.

Using the joint PDF from Problem 7:

Calculate the probability that the sum of wind speed and vibration amplitude is less than 1, i.e., $P(X + Y < 1)$.

The joint CDF of two continuous random variables $X$ and $Y$ is given by:

Find the probability that $X \le 1$ and $Y \le 2$, and derive the joint PDF $f(x,y)$.

Using the joint PDF derived in Problem 10:

Determine if the variables $X$ and $Y$ are independent.