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Chapter 4: Q124SE (page 197)

Consider an rv X with mean \({\rm{\mu }}\) and standard deviation \({\rm{\sigma }}\), and let g(X) be a specified function of X. The first-order Taylor series approximation to g(X) in the neighborhood of \({\rm{\mu }}\) is \({\rm{g(X)}} \approx {\rm{g(\mu ) + g'(\mu ) \times (X - \mu )}}\) The right-hand side of this equation is a linear function of X. If the distribution of X is concentrated in an interval over which is approximately linear (e.g., \(\sqrt {\rm{x}} \) is approximately linear in (1, 2)), then the equation yields approximations to E(g(X)) and V(g(X)).

a. Give expressions for these approximations. (Hint: Use rules of expected value and variance for a linear function \({\rm{aX + b}}\).)

b. If the voltage v across a medium is fixed but current, I is random, then resistance will also be a random variable related to I by \({\rm{R = v/I}}\). If \({{\rm{\mu }}_{\rm{I}}}{\rm{ = 20}}\) and \({{\rm{\sigma }}_{\rm{I}}}{\rm{ = }}{\rm{.5}}\), calculate approximations to \({{\rm{\mu }}_{\rm{R}}}\) and \({{\rm{\sigma }}_{\rm{R}}}\).

Short Answer

Expert verified

(a) \({\rm{E(g(x)) = g(\mu )}}\) and \({\rm{V(g(x)) = }}{\left( {{\rm{g'(\mu )}}} \right)^{\rm{2}}}{\rm{ \times }}{{\rm{\sigma }}^{\rm{2}}}\) is expressions for these approximately.

(b) \({{\rm{\mu }}_{\rm{R}}}{\rm{ = }}\frac{{\rm{v}}}{{{\rm{20}}}}\) and \({{\rm{\sigma }}_{\rm{R}}}{\rm{ = }}\frac{{\rm{v}}}{{{\rm{800}}}}\).

Step by step solution

01

Definition of Standard deviation

The standard deviation is a measurement of a set of values' variation or dispersion. A low standard deviation implies that the values are close to the set's mean (also known as the anticipated value), whereas a high standard deviation shows that the values are spread out over a larger range.

02

Determining expression for the approximation

(a) X is a rv with a mean of \({\rm{\mu }}\) and a standard deviation of \({\rm{\sigma }}\). By definition, we may then write

\({\rm{\mu = E(X)}}\)

\({{\rm{\sigma }}^{\rm{2}}}{\rm{ = E}}\left( {{{{\rm{(X - \mu )}}}^{\rm{2}}}} \right)\)

It's also assumed that g(X) is a defined function of X, with an approximation near \({\rm{\mu }}\) as:

\({\rm{g(X) = g(\mu ) + g'(\mu ) \times (X - \mu )}}\)

This expression is comparable to \({\rm{aX + b}}\) because \({\rm{g(\mu )}}\) and \({\rm{g'(\mu )}}\) are constants. The mean can thus be expressed as:

\({\rm{E(g(X)) = E(g(\mu )) + E}}\left( {{\rm{g'(\mu ) \times (X - \mu )}}} \right)\)

\({\rm{ = g(\mu ) + g'(\mu ) \times E((X - \mu ))}}\)

\({\rm{ = g(\mu ) + g'(\mu ) \times (E(X) - E(\mu ))}}\)

\({\rm{ = g(\mu ) + g'(\mu ) \times (\mu - \mu )}}\)

\({\rm{E(g(X)) = g(\mu )}}\)

03

Determining representation of g(X) variance

\({\rm{E}}\left( {{\rm{g(X}}{{\rm{)}}^{\rm{2}}}} \right){\rm{ = E}}\left( {{{\left( {{\rm{g(\mu ) + g'(\mu ) \times (X - \mu )}}} \right)}^{\rm{2}}}} \right)\)

\({\rm{ = E}}\left( {{{{\rm{(g(\mu ))}}}^{\rm{2}}}{\rm{ + 2 \times g(\mu ) \times g'(\mu ) \times (X - \mu ) + }}{{\left( {{\rm{g'(\mu )}}} \right)}^{\rm{2}}}{{{\rm{(X - \mu )}}}^{\rm{2}}}} \right)\)

\({\rm{ = E}}\left( {{{{\rm{(g(\mu ))}}}^{\rm{2}}}} \right){\rm{ + E}}\left( {{\rm{2 \times g(\mu ) \times g'(\mu ) \times (X - \mu )}}} \right){\rm{ + E}}\left( {{{\left( {{\rm{g'(\mu )}}} \right)}^{\rm{2}}}{{{\rm{(X - \mu )}}}^{\rm{2}}}} \right)\)

\({\rm{ = (g(\mu )}}{{\rm{)}}^{\rm{2}}}{\rm{ + 2 \times g(\mu ) \times g'(\mu ) \times E((X - \mu )) + }}{\left( {{\rm{g'(\mu )}}} \right)^{\rm{2}}}{\rm{E}}\left( {{{{\rm{(X - \mu )}}}^{\rm{2}}}} \right)\)

\({\rm{ = (g(\mu )}}{{\rm{)}}^{\rm{2}}}{\rm{ + 0 + }}{\left( {{\rm{g'(\mu )}}} \right)^{\rm{2}}}{\rm{ \times }}{{\rm{\sigma }}^{\rm{2}}}\)

\({\rm{E}}\left( {{\rm{g(X}}{{\rm{)}}^{\rm{2}}}} \right){\rm{ = (g(\mu )}}{{\rm{)}}^{\rm{2}}}{\rm{ + }}{\left( {{\rm{g'(\mu )}}} \right)^{\rm{2}}}{\rm{ \times }}{{\rm{\sigma }}^{\rm{2}}}\)

As a result, g(X) variance can be represented as:

\({\rm{V(g(x)) = E}}\left( {{\rm{g(X}}{{\rm{)}}^{\rm{2}}}} \right){\rm{ - E(g(X)}}{{\rm{)}}^{\rm{2}}}\)

\({\rm{ = }}\left( {{{{\rm{(g(\mu ))}}}^{\rm{2}}}{\rm{ + }}{{\left( {{\rm{g'(\mu )}}} \right)}^{\rm{2}}}{\rm{ \times }}{{\rm{\sigma }}^{\rm{2}}}} \right){\rm{ - (g(\mu )}}{{\rm{)}}^{\rm{2}}}\)

\({\rm{V(g(x)) = }}{\left( {{\rm{g'(\mu )}}} \right)^{\rm{2}}}{\rm{ \times }}{{\rm{\sigma }}^{\rm{2}}}\)

04

Calculating approximation for \({{\rm{\mu }}_{\rm{R}}}\) and  \({{\rm{\sigma }}_{\rm{R}}}\)

(b) Assume that I denote the current flowing through a medium, and that:

\({{\rm{\mu }}_{\rm{I}}}{\rm{ = 20}}\)

\({{\rm{\sigma }}_{\rm{I}}}{\rm{ = 0}}{\rm{.5}}\)

R represents the medium's resistance, and

\({\rm{R = }}\frac{{\rm{v}}}{{\rm{I}}}\)

I is equivalent to X, and R is equivalent to g(X), therefore

\({\rm{g(I) = R = }}\frac{{\rm{v}}}{{\rm{I}}}\)

\({\rm{g'(I) = }}\frac{{{\rm{ - v}}}}{{{{\rm{I}}^{\rm{2}}}}}\)

The mean and variance of R can be approximated using the data from the preceding section:

\({{\rm{\mu }}_{\rm{R}}}{\rm{ = g}}\left( {{{\rm{\mu }}_{\rm{I}}}} \right){\rm{ = g(20)}}\)

\({{\rm{\mu }}_{\rm{R}}}{\rm{ = }}\frac{{\rm{v}}}{{{\rm{20}}}}\)

\({{\rm{\sigma }}_{\rm{R}}}{\rm{ = g'(\mu ) \times \sigma = g'(20) \times (0}}{\rm{.5)}}\)

\({{\rm{\sigma }}_{\rm{R}}}{\rm{ = }}\frac{{\rm{v}}}{{{\rm{2}}{{\rm{0}}^{\rm{2}}}}}{\rm{ \times (0}}{\rm{.5)}}\)

\({{\rm{\sigma }}_{\rm{R}}}{\rm{ = }}\frac{{\rm{v}}}{{{\rm{800}}}}\)

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Most popular questions from this chapter

As in the case of the Weibull and Gamma distributions, the lognormal distribution can be modified by the introduction of a third parameter \({\rm{\gamma }}\) such that the pdf is shifted to be positive only for \({\rm{\chi > \gamma }}\)The article cited in Exercise \({\rm{4}}{\rm{.39}}\)suggested that a shifted lognormal distribution with shift (i.e., threshold) \({\rm{ = 1}}{\rm{.0}}\), mean value \({\rm{ = 2}}{\rm{.16, }}\), and standard deviation \({\rm{ = 1}}{\rm{.03}}\) would be an appropriate model for the \({\rm{rv X = }}\) maximum-to-average depth ratio of a corrosion defect in pressurized steel.

a. What are the values of \({\rm{\mu and \sigma }}\)for the proposed distribution?

b. What is the probability that depth ratio exceeds \({\rm{2}}\)?

c. What is the median of the depth ratio distribution?

d. What is the \({\rm{99th}}\) percentile of the depth ratio distribution?

Let \({\rm{X}}\) be a continuous \({\rm{rv}}\) with cdf

\({\rm{F(x) = }}\left\{ {\begin{array}{*{20}{c}}{\rm{0}}&{{\rm{x}} \le {\rm{0}}}\\{\frac{{\rm{x}}}{{\rm{4}}}\left( {{\rm{1 + ln}}\left( {\frac{{\rm{4}}}{{\rm{x}}}} \right)} \right)}&{{\rm{0 < x}} \le {\rm{4}}}\\{\rm{1}}&{{\rm{x > 4}}}\end{array}} \right.\)

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The lifetime \({\rm{X}}\) (in hundreds of hours) of a certain type of vacuum tube has a Weibull distribution with parameters \({\rm{\alpha = 2}}\)and \({\rm{\beta = 3}}\). Compute the following:

a. \({\rm{E(X)}}\) and \({\rm{V(X)}}\)

b. \(P(X \le 6)\)

c. \(P(1.5 \le X \le 6)\)

Two machines that produce wine corks, the first one having a normal diameter distribution with mean value \({\rm{3\;cm}}\) and standard deviation\({\rm{.1\;cm}}\), and the second having a normal diameter distribution with mean value \({\rm{3}}{\rm{.04\;cm}}\) and standard deviation\({\rm{.02\;cm}}\). Acceptable corks have diameters between \({\rm{2}}{\rm{.9}}\) and\({\rm{3}}{\rm{.1\;cm}}\). If \({\rm{60\% }}\) of all corks used come from the first machine and a randomly selected cork is found to be acceptable, what is the probability that it was produced by the first machine?

The article 鈥淎 Model of Pedestrians鈥� Waiting Times for Street Crossings at Signalized Intersections鈥� (Transportation Research, \({\rm{2013: 17--28}}\)) suggested that under some circumstances the distribution of waiting time X could be modelled with the following pdf:

\({\rm{f(x;\theta ,\tau ) = }}\left\{ {\begin{array}{*{20}{c}}{\frac{{\rm{\theta }}}{{\rm{\tau }}}{{{\rm{(1 - x/\tau )}}}^{{\rm{\theta - 1}}}}}&{{\rm{0}} \le {\rm{x < \tau }}}\\{\rm{0}}&{{\rm{ otherwise }}}\end{array}} \right.\)

a. Graph \({\rm{f(x;\theta ,80)}}\) for the three cases \({\rm{\theta = 4,1}}\) and \({\rm{.5}}\) (these graphs appear in the cited article) and comment on their shapes. b. Obtain the cumulative distribution function of X. c. Obtain an expression for the median of the waiting time distribution. d. For the case \({\rm{\theta = 4,\tau = 80}}\) calculate \({\rm{P(50}} \le {\rm{X}} \le {\rm{70)}}\) without at this point doing any additional integration.
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