Exercise 6: Random Variables and Probability
Distributions
Objective: Understandf random variables, distribution functions, density func-
tions, variance, and covariance.
3
1. Random Variables:
Define what a random variable is and differentiate between discrete
and continuous random variables. Provide one example for each type.
2. Distribution Function:
Given the discrete random variable X with the following probability
mass function (PMF):
$$P(X = x) = \begin{cases}
\frac{1}{4} & \text{for } x = 1, \\
\frac{1}{2} & \text{for } x = 2, \\
\frac{1}{4} & \text{for } x = 3.
\end{cases}$$
Compute the cumulative distribution function (CDF) F(x) for
X.
3. Density Function:
Let Y be a continuous random variable with the probability density
function (PDF):
$$f_Y(y) = \begin{cases}
\frac{3}{2}y & \text{for } 0 \le y \le 1, \\
0 & \text{otherwise}.
\end{cases}$$
Calculate the probability P(0.2 ≤ Y ≤ 0.6).
4. Variance and Covariance:
• Consider two random variables X and Y with the following joint
probability distribution:
$$P(X = x, Y = y) = \begin{cases}
\frac{1}{8} & \text{for } (x, y) = (1, 1), \\
\frac{1}{8} & \text{for } (x, y) = (1, 2), \\
\frac{1}{8} & \text{for } (x, y) = (2, 1), \\
\frac{1}{8} & \text{for } (x, y) = (2, 2), \\
0 & \text{otherwise}.
\end{cases}$$
Calculate the variance of X.
Calculate the variance of Y.
Compute the covariance of X and Y.
