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Let \(X=\) the hourly median power (in decibels) of received radio signals transmitted between two cities. The authors of the article "Families of Distributions for Hourly Median Power and Instantaneous Power of Received Radio Signals" (J. Research National Bureau of Standards, vol. 67D, 1963: 753-762) argue that the lognormal distribution provides a reasonable probability model for \(X\). If the parameter values are \(\mu=3.5\) and \(\sigma=1.2\), calculate the following: a. The mean value and standard deviation of received power. b. The probability that received power is between 50 and \(250 \mathrm{~dB} .\) c. The probability that \(X\) is less than its mean value. Why is this probability not \(.5\) ?

Step by step solution

01

Understand the Lognormal Distribution

If a random variable is log-normally distributed, then the natural logarithm of that variable is normally distributed. In this case, if \( X \) is log-normally distributed with parameters \( \mu = 3.5 \) and \( \sigma = 1.2 \), then \( \ln(X) \) is normally distributed with a mean of \( \mu \) and standard deviation of \( \sigma \).

02

Calculate the Mean of X

The mean of a lognormal distribution \( X \) with parameters \( \mu \) and \( \sigma \) can be calculated using the formula: \( E(X) = e^{\mu + \frac{\sigma^2}{2}} \). Substituting the given values, \( E(X) = e^{3.5 + \frac{1.2^2}{2}} = e^{3.5 + 0.72} = e^{4.22} \).

03

Calculate the Standard Deviation of X

The standard deviation of a lognormal distribution \( X \) is given by \( \sqrt{(e^{\sigma^2} - 1) \cdot e^{2\mu + \sigma^2}} \). Substituting the values: \( \sqrt{(e^{1.2^2} - 1) \cdot e^{2 \cdot 3.5 + 1.2^2}} \). First, calculate \( e^{1.44} \approx 4.22 \), then compute the entire expression.

04

Calculate the Probability that X is between 50 and 250

To find \( P(50 < X < 250) \), we need to convert \( X \) values to \( Z \)-scores using \( Z = \frac{\ln(X) - \mu}{\sigma} \). Calculate \( Z \) for 50 and 250 and use standard normal distribution tables or tools.

05

Calculate the Probability that X is Less Than Its Mean

Since \( X \) is log-normal, \( P(X < E(X)) = P(\ln(X) < \ln(E(X))) \). This can be calculated using the properties of the normal distribution. However, because the lognormal distribution is skewed, this probability is not 0.5 even though it might be intuitive to think it should be, due to the skewness of the lognormal distribution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Models

Probability models play a fundamental role in statistics, helping us understand and predict the behavior of complex systems. In the context of the exercise, the lognormal distribution is used as a probability model to describe the hourly median power of radio signals, denoted by the variable \( X \).
Lognormal distributions are particularly useful when modeling variables that are positively skewed, non-negative, and multiplicative in nature, making them suitable for financial, biological, and environmental data.

• To use a lognormal model, we take advantage of its connection to the normal distribution. The natural logarithm of a lognormally distributed variable is normally distributed, which provides a bridge to use the properties of the normal distribution.
• The relevance of using a lognormal distribution as the probability model in this case is due to the nature of the data: the power levels exhibit positive skewness and are non-negative.
• By understanding that \( \ln(X) \) is normally distributed, we can apply various statistical methods to derive meaningful insights about the original variable \( X \).

Mean and Standard Deviation

The mean and standard deviation are core statistical measures used to describe the central tendency and spread respectively. When dealing with a lognormal distribution for a variable \( X \), these measures behave differently than in a normal distribution.

• The mean of a lognormal distribution is calculated using the formula: \( E(X) = e^{\mu + \frac{\sigma^2}{2}} \). This reflects how the parameters \( \mu \) and \( \sigma \), which are the mean and standard deviation of \( \ln(X) \), influence the mean of \( X \).
• In the exercise, with given parameters \( \mu = 3.5 \) and \( \sigma = 1.2 \), the mean is computed as \( E(X) = e^{4.22} \).
• Standard deviation, describing how spread out the values are, is calculated as \( \sqrt{(e^{\sigma^2} - 1) \cdot e^{2\mu + \sigma^2}} \). This accounts for the multiplicative nature of the lognormal variable.
• These calculations provide key insights into the expected values and variability of the radio signal power, which are crucial for reliability assessments.

Skewness in Distributions

Skewness is a measure of asymmetry or distortion from the symmetric bell curve in a data distribution. In the case of the lognormal distribution, skewness is always positive, reflecting a right-skewed distribution that indicates a longer tail on the right.

• This skewness significantly impacts probabilities associated with \( X \). For example, the probability that \( X \) is less than its mean is not 0.5 as you might expect in a symmetric distribution like the normal distribution.
• In the original exercise, the probability of \( X \) being less than its mean is less than 0.5 due to this asymmetry. Most values are concentrated just below the mode, with the mean pulled rightward by higher values in the tail.

Understanding skewness helps clarify why relying solely on the mean for interpreting lognormal distributions might be misleading without considering the distribution shape.

Normal Distribution

The normal distribution, often called a Gaussian distribution, is a fundamental concept in statistics and is characterized by its symmetric bell-shaped curve. It serves as a basis for the lognormal distribution, where the natural logarithm of lognormally distributed variables is normally distributed.

• The properties of the normal distribution, such as its symmetry and defined probabilities, are utilized in analyzing lognormal variables. For example, we convert lognormal values to \( Z \)-scores to find probabilities using the standard normal distribution.
• In the exercise, to determine the probability that the signal power \( X \) falls between specific values or is less than a threshold, we make use of the normal distribution characteristics.
• Converting lognormal data back and forth with the normal distribution allows us to apply well-established techniques to infer probability levels and other statistical properties.

Thus, understanding the connection between the two distributions is crucial for effectively using lognormal models in statistical analysis and predictions.