Problem 12 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

Consider the independent random variables \(H\) and \(R\) from the vase example, with a \(U(25,35)\) and a \(U(7.5,12.5)\) distribution. Compute \(\mathrm{E}[H]\) and \(\mathrm{E}\left[R^{2}\right]\) and check that \(\mathrm{E}[V]=\pi \mathrm{E}[H] \mathrm{E}\left[R^{2}\right]\).

Step-by-Step Solution

Answer

\(\mathrm{E}[V] = \pi \cdot 30 \cdot 102.0833\).

Step 1: Calculate \(\mathrm{E}[H]\)

The random variable \(H\) follows a uniform distribution \(U(25,35)\). The expected value for a uniform distribution \(U(a,b)\) is given by \(\frac{a+b}{2}\).\[\mathrm{E}[H] = \frac{25 + 35}{2} = 30\]

Step 2: Calculate \(\mathrm{E}[R^2]\)

The random variable \(R\) follows a uniform distribution \(U(7.5,12.5)\). For uniform random variables, the expected value of the square is given by \(\mathrm{E}[X^2] = \frac{b^3-a^3}{3(b-a)}\) if \(X\sim U(a,b)\). So:\[\mathrm{E}[R^2] = \frac{12.5^3 - 7.5^3}{3(12.5-7.5)}\]Calculate the cubes and substitute:\[\mathrm{E}[R^2] = \frac{1953.125 - 421.875}{15} = 102.08333333\]

Step 3: Compute \(\mathrm{E}[V]\) and check equality

Compute \(\mathrm{E}[V] = \pi \mathrm{E}[H] \mathrm{E}[R^2]\):\[\mathrm{E}[V] = \pi \cdot 30 \cdot 102.08333333 \] Approximating \(\pi\) as 3.1416, calculate:\[\mathrm{E}[V] = 3.1416 \times 30 \times 102.08333333 \approx 9625.1635\]

Key Concepts

Uniform Distribution

Uniform distribution is a type of probability distribution where all outcomes are equally likely within a given interval. In this type of distribution, every number between the lower limit \( a \) and the upper limit \( b \) has an equal chance of occurring. This makes it an ideal model for scenarios where each result is naturally balanced or neutral.

In mathematical terms, if a random variable \( X \) follows a uniform distribution over the interval \( [a, b] \), it is typically denoted as \( U(a, b) \). The expected value, or the mean of a uniform distribution, is calculated by averaging the boundaries:\[\mathrm{E}[X] = \frac{a+b}{2}\]
For the random variable \( H \) following a uniform distribution \( U(25,35) \), the calculation is straightforward:

• Lower limit \( a = 25 \)
  
• Upper limit \( b = 35 \)
  

By applying the expected value formula for uniform distribution, we determine that \( \mathrm{E}[H] = 30 \).
Uniform distributions are applicable in many fields, including quality control and gaming strategies, where each outcome is equally possible.

Independent Random Variables

Independent random variables are an essential concept in probability and statistics. Two random variables \( X \) and \( Y \) are independent if the occurrence of one does not affect the occurrence of the other. Mathematically, this can be expressed as:

• \( P(X \cap Y) = P(X) \cdot P(Y) \)
  

Independence allows us to calculate joint probabilities, expectations, and variances conveniently.

In the exercise, the variables \( H \) and \( R \) are independent. This implies that their expected values and other characteristics, like variance, are not influenced by each other. This property simplifies calculations such as finding expectations of functions involving both variables, as shown when calculating the expected value \( \mathrm{E}[V] = \pi \mathrm{E}[H] \mathrm{E}[R²] \).
Understanding independence is crucial for analyzing complex systems as it allows us to deconstruct problems into simpler, manageable parts.

Probability Calculations

Probability calculations are the technical heart of probability theory, encompassing the determination of the likelihood of various outcomes and events. These calculations often involve expected values, variances, and probabilities of specific events.

In the context of our exercise, probability calculations are used to determine the expected value of specific random variables and their transformations. For instance, the expected value of \( R^2 \) given \( R \sim U(7.5,12.5) \) is computed using:

• \( \mathrm{E}[X^2] = \frac{b^3 - a^3}{3(b-a)} \)
  

The computation involves numeric substitution and manipulation to derive\( \mathrm{E}[R^2] = 102.08333333 \), which is an essential step in determining \( \mathrm{E}[V] \).
Through these precise calculations, we gain insights into various real-world and theoretical scenarios. Mastery over these calculations empowers us to describe and predict outcomes accurately.