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Chapter 4: Problem 111

The mode of a continuous distribution is value \(x^{*}\) that maximizes \(f(x)\). a. What is the mode of a normal distribution with parameters \(\mu\) and \(\sigma\) ? b. Does the uniform distribution with parameters \(A\) and \(B\) have a single mode? Why or why not? c. What is the mode of an exponential distribution with parameter \(\lambda\) ? (Draw a picture.) d. If \(X\) has a gamma distribution with parameters \(\alpha\) and \(\beta\), and \(\alpha>1\), find the mode. e. What is the mode of a chi-squared distribution having \(\nu\) degrees of freedom?

Short Answer

Expert verified
a. The mode is \(\mu\). b. No, it does not have a single mode. c. The mode is at \(x = 0\). d. The mode is \((\alpha - 1)\beta\). e. The mode is \(\nu - 2\).

Step by step solution

01

Normal Distribution Mode

The mode of a continuous distribution is the value of \( x \) that maximizes the probability density function \( f(x) \). For a normal distribution with mean \( \mu \) and standard deviation \( \sigma \), the probability density function is symmetric around the mean. Therefore, the mode of the normal distribution is \( \mu \), as this is the peak of the distribution.
02

Uniform Distribution Mode

A uniform distribution with parameters \( A \) and \( B \) is constant between \( A \) and \( B \) and zero outside this interval. Therefore, every point between \( A \) and \( B \) has the same probability density value, meaning that there is no single value \( x \) that maximizes \( f(x) \). Hence, the uniform distribution does not have a single mode.
03

Exponential Distribution Mode

The probability density function of an exponential distribution is \( f(x) = \lambda e^{-\lambda x} \) for \( x \geq 0 \). Taking the derivative and setting it to zero to find the maximum, we get the mode at \( x = 0 \). This is because the exponential distribution decreases continuously from its maximum at \( x = 0 \).
04

Gamma Distribution Mode

For a gamma distribution with parameters \( \alpha \) and \( \beta \), the mode can be calculated using the formula \( (\alpha - 1)\beta \) when \( \alpha > 1 \). This is derived from solving the derivative of the gamma probability density function to find its peak.
05

Chi-Squared Distribution Mode

The chi-squared distribution is a special case of the gamma distribution where \( \alpha = \frac{u}{2} \) and \( \beta = 2 \). Using the gamma distribution mode formula, the mode for the chi-squared distribution is \( (\frac{u}{2} - 1) \cdot 2 = u - 2 \), provided \( u > 2 \).

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Key Concepts

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

Normal Distribution
A normal distribution, often referred to as a Gaussian distribution, is widely recognized by its symmetric bell-shaped curve. This distribution is defined by two parameters: the mean \( \mu \) and the standard deviation \( \sigma \). The mean represents the center of the distribution, where the data is most concentrated. In terms of probability, the mode of a distribution is the value of \( x \) that maximizes its probability density function (PDF). For a normal distribution, because of its symmetry, the highest point of the curve, which is also the mode, is located exactly at the mean \( \mu \). This means that the normal distribution has a single mode at the mean.
Uniform Distribution
The uniform distribution, in its simplest form, refers to a distribution where all outcomes are equally likely within a defined interval. It is described by two parameters, \( A \) and \( B \), which set the bounds of this interval. Within this range, the distribution is flat, meaning every point has an identical likelihood. The value of the probability density function remains constant between \( A \) and \( B \). Consequently, there is no single mode because no single value is preferred over another. Thus, a uniform distribution does not have a unique mode, reflecting its equal distribution of probabilities across its range.
Exponential Distribution
The exponential distribution is characterized by a rapid decline in probabilities as \( x \) increases, making it unique compared to others. Its probability density function is given by \( f(x) = \lambda e^{-\lambda x} \) for \( x \geq 0 \), where \( \lambda \) is the rate parameter. To find its mode, we consider the point where this function reaches its peak. By taking the derivative of \( f(x) \) and setting it to zero, we find that the mode occurs at \( x = 0 \). This signifies that the probability is at its maximum at the beginning of the distribution and decreases swiftly from there.
Gamma Distribution
The gamma distribution is a flexible distribution characterized by two parameters: shape \( \alpha \) and scale \( \beta \). It is commonly used to model waiting times or life data. The mode of the gamma distribution can be found using the formula \( (\alpha - 1)\beta \), provided that \( \alpha > 1 \). This formula indicates that the mode depends heavily on the values of \( \alpha \) and \( \beta \), shifting the peak of the distribution as these parameters change. For \( \alpha \leq 1 \), the mode exists at zero due to the nature of the formula.
Chi-Squared Distribution
The chi-squared distribution is a specific type of gamma distribution. It emerges when analyzing data based on the goodness of fit, variances, and more. It is defined by the parameter \( u \), which represents the degrees of freedom. For the chi-squared distribution, the mode is calculated using the expression \( u - 2 \), assuming \( u > 2 \). This connects to its relation with the gamma distribution, where \( \alpha = \frac{u}{2} \) and \( \beta = 2 \). Consequently, just like with the gamma distribution, the degrees of freedom determine the positioning of the mode within the distribution.

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Most popular questions from this chapter

The article "On Assessing the Accuracy of Offshore Wind Turbine Reliability- Based Design Loads from the Environmental Contour Method" (Intl. J. of Offshore and Polar Engr., 2005: 132-140) proposes the Weibull distribution with \(\alpha=1.817\) and \(\beta=.863\) as a model for 1-hour significant wave height ( \(m\) ) at a certain site. a. What is the probability that wave height is at most \(.5 \mathrm{~m}\) ? b. What is the probability that wave height exceeds its mean value by more than one standard deviation? c. What is the median of the wave-height distribution? d. For \(0

Let \(X\) have the Pareto pdf $$ f(x ; k, \theta)=\left\\{\begin{array}{cc} \frac{k \cdot \theta^{k}}{x^{k+1}} & x \geq \theta \\ 0 & x<\theta \end{array}\right. $$ introduced in Exercise \(10 .\) a. If \(k>1\), compute \(E(X)\). b. What can you say about \(E(X)\) if \(k=1\) ? c. If \(k>2\), show that \(V(X)=k \theta^{2}(k-1)^{-2}(k-2)^{-1}\). d. If \(k=2\), what can you say about \(V(X)\) ? e. What conditions on \(k\) are necessary to ensure that \(E\left(X^{n}\right)\) is finite?

Let \(X\) denote the amount of time a book on two-hour reserve is actually checked out, and suppose the cdf is $$ F(x)= \begin{cases}0 & x<0 \\ \frac{x^{2}}{4} & 0 \leq x<2 \\ 1 & 2 \leq x\end{cases} $$ a. Calculate \(P(X \leq 1)\). b. Calculate \(P(.5 \leq X \leq 1)\). c. Calculate \(P(X>1.5)\). d. What is the median checkout duration \(\tilde{\mu}\) ? [solve \(.5=F(\widetilde{\mu})]\). e. Obtain the density function \(f(x)\). f. Calculate \(E(X)\). g. Calculate \(V(X)\) and \(\sigma_{X}\). h. If the borrower is charged an amount \(h(X)=X^{2}\) when checkout duration is \(X\), compute the expected charge \(E[h(X)]\).

An oocyte is a female germ cell involved in reproduction. Based on analyses of a large sample, the article "'Reproductive Traits of Pioneer Gastropod Species Colonizing Deep-Sea Hydrothermal Vents After an Eruption" (Marine Biology, 2011: 181-192) proposed the following mixture of normal distributions as a model for the distribution of \(X=\) oocyte diameter \((\mu \mathrm{m})\) : $$ f(x)=p f_{1}\left(x ; \mu_{1}, \sigma\right)+(1-p) f_{2}\left(x ; \mu_{2}, \sigma\right) $$ where \(f_{1}\) and \(f_{2}\) are normal pdfs. Suggested parameter values were \(p=.35, \mu_{1}=4.4, \mu_{2}=5.0\), and \(\sigma=.27\). a. What is the expected (i.e. mean) value of oocyte diameter? b. What is the probability that oocyte diameter is between \(4.4 \mu \mathrm{m}\) and \(5.0 \mu \mathrm{m}\) ? c. What is the probability that oocyte diameter is smaller than its mean value? What does this imply about the shape of the density curve?

Let \(X\) denote the distance (m) that an animal moves from its birth site to the first territorial vacancy it encounters. Suppose that for banner-tailed kangaroo rats, \(X\) has an exponential distribution with parameter \(\lambda=.01386\) (as suggested in the article "Competition and Dispersal from Multiple Nests," Ecology, 1997: 873鈥�883). a. What is the probability that the distance is at most \(100 \mathrm{~m}\) ? At most \(200 \mathrm{~m}\) ? Between 100 and \(200 \mathrm{~m}\) ? b. What is the probability that distance exceeds the mean distance by more than 2 standard deviations? c. What is the value of the median distance?
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