Problem 172 - Applied Statistics and Probability for Engineers

Question

Suppose that \(X\) has a lognormal distribution with parameters \(\theta=2\) and \(\omega^{2}=4 .\) Determine the following in parts (a) and (b): (a) \(P(X<500)\) (b) Conditional probability that \(X<1500\) given that \(X>1000\) (c) What does the difference between the probabilities in parts (a) and (b) imply about lifetimes of lognormal random variables?

Step-by-Step Solution

Answer

(a) 0.9826; (b) 0.4583; (c) Indicates decreasing likelihood to reach higher values once past a threshold.

Step 1: Understanding the Lognormal Distribution

The lognormal distribution assumes that if a random variable \( X \) is lognormally distributed, then \( Y = \ln(X) \) is normally distributed. Given \( \theta = 2 \) and \( \omega^2 = 4 \), \( Y \) is normally distributed with mean \( \theta = 2 \) and variance \( \omega^2 = 4 \). Therefore, \( Y \sim N(2, 4) \).

Step 2: Transforming Probability for Part (a)

To find \( P(X<500) \), we first transform it to the normal distribution. We need \( P(\ln(X) < \ln(500)) \). This is equivalent to finding \( P(Y < \ln(500)) \).

Step 3: Calculating \( \ln(500) \)

Compute \( \ln(500) \) to find the equivalent normal probability. \( \ln(500) \approx 6.2146 \).

Step 4: Standardizing Normal Distribution for Part (a)

Standardize using \( Z = \frac{Y - 2}{2} \). Now find \( P\left( \frac{Y - 2}{2} < \frac{6.2146 - 2}{2} \right) = P(Z < 2.1073) \).

Step 5: Finding Probability for Part (a)

Use the standard normal distribution table to find \( P(Z < 2.1073) \), which is approximately \( 0.9826 \). Thus, \( P(X<500) \approx 0.9826 \).

Step 6: Finding Probability for Part (b) - Conditional Probability

For \( P(X<1500 \mid X>1000) \), express it as \( \frac{P(1000 < X < 1500)}{P(X>1000)} \).

Step 7: Transforming Interval Probability to Normal Distribution

Find \( P(\ln(1000) < Y < \ln(1500)) \). Compute \( \ln(1000) \approx 6.9078 \) and \( \ln(1500) \approx 7.3132 \).

Step 8: Standardizing Normal Distribution for Interval

Calculate \( P\left( \frac{6.9078 - 2}{2} < Z < \frac{7.3132 - 2}{2} \right) = P(2.4543 < Z < 2.6566) \).

Step 9: Finding Interval Probabilities

Using the standard normal distribution, \( P(Z < 2.6566) \approx 0.9961 \) and \( P(Z < 2.4543) \approx 0.9928 \). The interval probability \( P(1000 < X < 1500) \approx 0.9961 - 0.9928 = 0.0033 \).

Step 10: Finding Probability Component for Denominator

For \( P(X>1000) \), use \( 1 - P(X<1000) \). Compute \( P(Y < 6.9078) = P(Z < 2.454) \approx 0.9928 \), so \( P(X>1000) \approx 0.0072 \).

Step 11: Calculate Conditional Probability for Part (b)

Substituting in the conditional probability formula, \( P(X<1500 \mid X>1000) = \frac{0.0033}{0.0072} \approx 0.4583 \).

Step 12: Analyzing Probabilities for Part (c)

Compare the probabilities. \( P(X<500) = 0.9826 \) vs \( P(X<1500 \mid X>1000) = 0.4583 \). This suggests that once a lognormal random variable exceeds a certain threshold, its additional probability to reach higher values reduces significantly.

Key Concepts

Probability Theory

Probability theory is the mathematical framework that allows us to analyze the likelihood of different events. It provides the tools to measure the occurrence, or non-occurrence, of various outcomes in a random experiment. This framework is fundamental because it helps us understand random phenomena, which occur frequently in various fields such as finance, science, and engineering.
To utilize probability theory, one begins with the concept of a probability space. This includes:

• Sample Space (\( S \)): The set of all possible outcomes of an experiment.
• Events: Subsets of the sample space that we are interested in.
• Probability Function: A rule that assigns each event a probability, denoted as \( P(A) \), which falls between 0 and 1.

The sum of probabilities of all possible events always equals 1. It helps us make sense of uncertainty by providing a numeric measure of how likely an event is to occur. As we delve into specific distributions like the lognormal, probability theory provides the foundational principles we apply to solve problems and interpret data.

Conditional Probability

Conditional probability represents the likelihood of an event occurring given that another event has already taken place. It's a crucial concept in probability theory because it helps refine predictions based on additional known information. When dealing with conditional probability, we focus on a particular part of the sample space that meets the condition.
For two events, A and B, the conditional probability of A given that B has occurred is written as \( P(A \,|\, B) \). Calculated using the formula: \[ P(A \,|\, B) = \frac{P(A \cap B)}{P(B)} \]When handling problems like finding the probability of \( X < 1500 \) given \( X > 1000 \) in a lognormal distribution, the conditional probability helps us assess how likely we've observed values in a particular interval given some known range. It requires calculating the joint probability and dividing it by the probability of the known condition. This process allows us to adjust our understanding of probabilities in scenarios where some information is already revealed.

Standard Normal Distribution

The standard normal distribution is a crucial statistical tool used for standardizing values from any normal distribution. It's defined as a normal distribution with a mean of 0 and a standard deviation of 1. This concept is essential when working with transformed variables, like those encountered in problems involving the lognormal distribution.
When you have a normal distribution \( Y \sim N(\theta, \omega^{2}) \), you can convert it to the standard normal distribution \( Z \) with the transformation: \[ Z = \frac{Y - \theta}{\omega} \]This standardization simplifies the process of calculating probabilities, as it allows us to use pre-calculated tables.
For example, in a problem where \( Y \sim N(2, 4) \), you transform a value like \( \ln(500) \) to a Z-score: \( Z = \frac{6.2146 - 2}{2} = 2.1073 \). Then, using the standard normal tables, you find the probability corresponding to this Z-score. By utilizing the standard normal distribution, we avoid the complexity of having to calculate areas under a different curve every time.

Random Variables

Random variables are core components of probability theory. They are variables whose possible values are numerical outcomes of a random phenomenon. Random variables allow us to quantify randomness and create models that represent real-world situations. There are two main types of random variables:

• Discrete Random Variables: Have distinct, separate values (like the result of a dice roll).
• Continuous Random Variables: Can take any value within a range, often involving intervals (like the height of students).

For instance, in the context of a lognormal distribution, random variable \( X \) might represent financial values, where its logarithm is normally distributed. The nature of random variables allows us to build distributions that can predict outcomes, like the probability of \( X < 500 \).
By manipulating random variables, we calculate possibilities and make inferences about larger populations based on the information captured in our variable's distribution. They are essential for modeling, analyzing, and interpreting data in a structured way.