Problem 2 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

We simulate the random variable \(X=1+2 \sqrt{U}\) constructed in Quick exercise 6.3. As realization for \(U\) we obtain from the pseudo random generator the number \(u=0.3782739 .\) a. What is the corresponding realization \(x\) of the random variable \(X ?\) b. If the next call to the random generator yields \(u=0.3\), will the corresponding realization for \(X\) be larger or smaller than the value you found in \(\mathbf{a}\) ? c. What is the probability the next draw will be smaller than the value you found in \(\mathbf{a}\) ?

Step-by-Step Solution

Answer

a. \(x \approx 2.2312\); b. Smaller; c. 0.3782.

Step 1: Understanding the Random Variable Transformation

The random variable \(X\) is defined as \(X = 1 + 2\sqrt{U}\), where \(U\) is a uniform random variable on the interval [0,1]. We need to find the value of \(x\) when \(u=0.3782739\).

Step 2: Calculate the Realization of \(X\) for \(u = 0.3782739\)

Substitute \(u = 0.3782739\) into the transformation equation: \[ x = 1 + 2\sqrt{0.3782739} \]. First, calculate \(\sqrt{0.3782739}\), which is approximately \(0.6156\). Then compute the overall value: \[ x = 1 + 2 \times 0.6156 = 2.2312 \].

Step 3: Calculate the Realization of \(X\) for Next \(u = 0.3\)

Use the same formula for \(X\) with the new \(u = 0.3\): \[ x = 1 + 2\sqrt{0.3} \]. Calculate \(\sqrt{0.3}\), which is approximately \(0.5477\). Then compute: \[ x = 1 + 2 \times 0.5477 = 2.0954 \]. This value is smaller than the \(x\) computed in Step 2.

Step 4: Probability of Next Draw Being Smaller

Since \(X = 1 + 2\sqrt{U}\), we need the probability that \(X < 2.2312\) when \(U\sim \text{Uniform}(0,1)\). Rearrange to find \(U\): \[ 1 + 2\sqrt{U} < 2.2312 \quad \Rightarrow \quad 2\sqrt{U} < 1.2312 \quad \Rightarrow \quad \sqrt{U} < 0.6156 \]. Squaring both sides, we obtain \(U < 0.3782\). The probability of \(U < 0.3782\) for a uniform distribution on [0,1] is 0.3782.

Key Concepts

Uniform Distribution

A uniform distribution is one of the simplest types of probability distributions. It means every value in a certain interval is equally likely to occur. For example, a random variable with a uniform distribution on the interval [0,1] has an equal chance of taking on any value in this range.
Understanding this gives us an essential foundation for simulating random variables like the one in the exercise. The random variable we deal with has transformation based on this uniform distribution. It is helpful to think of it as distributing values 'evenly' across an interval.

• The function is defined evenly across all points in the interval, where the total probability sums up to 1.
• This property makes it a popular choice for pseudo random number generators.

The ability to derive probabilities and expected outcomes from this foundation is a vital skill in probability and statistics.

Probability Calculation

Calculating probability in scenarios involving random variable transformations can be straightforward once you know the distribution of the original variable. For a uniform distribution, as in our exercise, calculating the probability of outcomes is particularly straightforward.
In Step 4, the task was to find the probability that a future realization of the uniform random variable will be smaller than a previously calculated value. This involved rearranging and solving the transformation equation to find a condition on the variable U.

• Understanding how probabilities map from one distribution to another is key.
• Simple algebra helps solve for thresholds, such as finding the probability that a draw will be smaller.
• Probabilities for uniform distributions along [0, 1] are simply the length of the interval.

This makes uniform distribution a great educational tool for teaching foundational probability concepts.

Pseudo Random Generator

A pseudo random generator is a type of algorithm used in computing to generate a sequence of numbers that approximates the properties of random numbers. These sequences are not truly random. But for practical purposes, they are sufficiently "random" for most tasks.
In the exercise, the pseudo random generator provides values for the uniform random variable U. These values are crucial as inputs for transformations that yield realizations of other random variables.

• Pseudo random generators are deterministic; given the same initial conditions or seeds, they will generate the same sequence of numbers.
• They are commonly used when true randomness is difficult or unnecessary to achieve.

Understanding how algorithms generate such sequences assists in fields ranging from cryptography to statistical simulations.

Realization of Random Variables

The realization of a random variable refers to the actual value it takes when observed. In probability theory, understanding realizations is crucial in relating abstract probability models to observable events.
When observing a random variable, such as in the problem discussed, the value you calculate after undergoing a transformation is a 'realization.'

• A realization can be thought of as a single outcome from the possible range of a random variable.
• When we use the transformation formula to find the value of X, using the generator's output for U, we are finding the realization of X.

Realizations help in comparing theoretical predictions with observed data, bridging the gap between mathematical theory and reality.