Questions and Answers of Applied Statistics And Probability For Engineers

$X$ is the number of bits in error in the next four bits transmitted. What is the expected value of the square of the number of bits in error? Now, $h(X)=X^{2}$. Therefore,\[ \begin{aligned}
In Example 4.1, $X$ is the current measured in milliamperes. What is the expected value of power when the resistance is 100 ohms?
Determine the probability mass function of $X$ from the following cumulative distribution function:\[ F(x)=\left\{\begin{array}{lr} 0 & x. \]Figure 3.3 displays a plot of $F(x)$. From the
Two new product designs are to be compared on the basis of revenue potential. Marketing believes that the revenue from design A can be predicted quite accurately to be $\$ 3$ million. The revenue
The probability that a wafer contains a large particle of contamination is 0.01. If it is assumed that the wafers are independent, what is the probability that exactly 125 wafers need to be analyzed
Consider the time to recharge the flash. The probability that a camera passes the test is 0.8, and the cameras perform independently. What is the probability that the third failure is obtained in
Let the continuous random variable $X$ denote the current measured in a thin copper wire in milliamperes. Assume that the range of $X$ is $[4.9,5.1] \mathrm{mA}$, and assume that the
The time until a chemical reaction is complete (in milliseconds) is approximated by the cumulative distribution function\[ F(x)= \begin{cases}0 & x
Assume that the current measurements in a strip of wire follow a normal distribution with a mean of 10 milliamperes and a variance of 4 (milliamperes) ${ }^{2}$. What is the probability that a
Suppose that the current measurements in a strip of wire are assumed to follow a normal distribution with a mean of 10 milliamperes and a variance of 4 (milliamperes) ${ }^{2}$. What is the
Assume that in a digital communication channel, the number of bits received in error can be modeled by a binomial random variable, and assume that the probability that a bit is received in error is
Assume that the number of asbestos particles in a squared meter of dust on a surface follows a Poisson distribution with a mean of 1000 . If a squared meter of dust is analyzed, what is the
In a large corporate computer network, user log-ons to the system can be modeled as a Poisson process with a mean of 25 log-ons per hour. What is the probability that there are no log-ons in an
Let $X$ denote the time between detections of a particle with a Geiger counter and assume that $X$ has an exponential distribution with $E(X)=1.4$ minutes. The probability that we detect a
The time to prepare a slide for high-throughput genomics is a Poisson process with a mean of two hours per slide. What is the probability that 10 slides require more than 25 hours to prepare?
A semiconductor product consists of three layers. Suppose that the variances in thickness of the first, second, and third layers are 25,40 , and 30 square nanometers, respectively, and the layer
Consider an experiment that selects a cell phone camera and records the recycle time of a flash (the time taken to ready the camera for another flash). The possible values for this time depend on the
Suppose that the recycle times of two cameras are recorded. The extension of the positive real line $R$ is to take the sample space to be the positive quadrant of the plane\[ S=R^{+} \times R^{+} \]
Each message in a digital communication system is classified as to whether it is received within the time specified by the system design. If three messages are classified, use a tree diagram to
Consider the sample space $S=\{y y, y n, n y, n n\}$ in Example 2.2. Suppose that the subset of outcomes for which at least one camera conforms is denoted as $E_{1}$. Then,\[ E_{1}=\{y y, y n, n
As in Example 2.1, camera recycle times might use the sample space $S=R^{+}$, the set of positive real numbers. Let\[ E_{1}=\{x \mid 10 \leq x
The following table summarizes visits to emergency departments at four hospitals in Arizona. People may leave without being seen by a physician, and those visits are denoted as LWBS. The remaining
The design for a Website is to consist of four colors, three fonts, and three positions for an image. From the multiplication rule, $4 \times 3 \times 3=36$ different designs are possible.
A printed circuit board has eight different locations in which a component can be placed. If four different components are to be placed on the board, how many different designs are possible?
A hospital operating room needs to schedule three knee surgeries and two hip surgeries in a day. We denote a knee and hip surgery as $k$ and $h$, respectively. The number of possible sequences of
A printed circuit board has eight different locations in which a component can be placed. If five identical components are to be placed on the board, how many different designs are possible?
A bin of 50 manufactured parts contains 3 defective parts and 47 nondefective parts. A sample of 6 parts is selected without replacement. That is, each part can be selected only once, and the sample
Assume that $30 \%$ of the laser diodes in a batch of 100 meet the minimum power requirements of a specific customer. If a laser diode is selected randomly, that is, each laser diode is equally
A random experiment can result in one of the outcomes $\{a, b, c, d\}$ with probabilities $0.1,0.3,0.5$, and 0.1 , respectively. Let $A$ denote the event $\{a, b\}, B$ the event \(\{b, c,
Consider the inspection described in Example 2.11. From a bin of 50 parts, 6 parts are selected randomly without replacement. The bin contains 3 defective parts and 47 nondefective parts. What is the
Table 2.1 lists the history of 940 wafers in a semiconductor manufacturing process. Suppose that 1 wafer is selected at random. Let $H$ denote the event that the wafer contains high levels of
Here is a simple example of mutually exclusive events, which are used quite frequently. Let $X$ denote the $\mathrm{pH}$ of a sample. Consider the event that $X$ is greater than 6.5 but less
Table 2.2 provides an example of 400 parts classified by surface flaws and as (functionally) defective. For this table, the conditional probabilities match those discussed previously in this section.
Again consider the 400 parts in Table 2.2. From this table,\[ P(D \mid F)=\frac{P(D \cap F)}{P(F)}=\frac{10}{400} / \frac{40}{400}=\frac{10}{40} \]Note that in this example all four of the following
The probability that the first stage of a numerically controlled machining operation for high-rpm pistons meets specifications is 0.90 . Failures are due to metal variations, fixture alignment,
Consider the contamination discussion at the start of this section. The information is summarized here. Probability of Failure Level of Probability of Contamination Level 0.1 High 0.2 0.005 Not high
Consider the inspection described in Example 2.11. Six parts are selected randomly from a bin of 50 parts, but assume that the selected part is replaced before the next one is selected. The bin
Consider the inspection described in Example 2.11. Six parts are selected randomly without replacement from a bin of 50 parts. The bin contains 3 defective parts and 47 nondefective parts. Let $A$
Reconsider Example 2.20. The conditional probability that a high level of contamination was present when a failure occurred is to be determined. The information from Example 2.20 is summarized
Let $D$ denote the event that you have the illness, and let $S$ denote the event that the test signals positive. The probability requested can be denoted as $P(D \mid S)$. The probability that
The simplest sequential description of the Ewens sampling formula is called the Chinese restaurant process. The first customer arrives and is seated at a table. After $n$ customers have been
According to the standard definition in Sect. 11.4.2, two observational units $u, u^{\prime}$ belong to the same experimental unit if the treatment assignment probabilities given the baseline
According to Villa et al.,Pigeons combat feather lice by removing them with their beaks during regular bouts of preening. Columbicola columbae, a parasite of feral pigeons, avoids preening by hiding
Explain why (5.3) is in conflict with randomization.
Use regress (. . . ) to compute the REML estimate of the variance components in (5.4). Hence obtain the estimated slopes, their difference, and the standard errors for all three.
Regress the $32 \times 9$ lineage-time averages (for PC1) against sample size using sample size as weights. You should find a statistically significant positive coefficient a little larger than
Given the variance components, the Bayes estimate of the secular trend is a linear combination of the fitted mean vector and the fitted residual\[\tilde{\mu}=P Y+L \Sigma^{-1} Q Y \text {, }\]where
For the cubic and quadratic models described in the preceding exercise, compute the predicted temperature for next year, i.e., the conditional distribution of the mean temperature for next year given
This exercise is concerned with two versions of the Bayes estimate of the seasonal rainfall component, where it is required to compute $E\left(\eta_{m} \midight.$ data) for each of 12 months. As
For the Oxford rainfall data up to Dec 2019, the first Bayes estimate in the preceding exercise is a flat $10 \%$ shrinkage of monthly averages towards the annual average; the second Bayes estimate
In this exercise, $\chi$ is the chordal metric on the unit circle. From the results of the preceding exercise, show that $4 / \pi-\chi\left(t, t^{\prime}ight)$ is positive definite on \([0,2
Show that the quadratic function\[\frac{2 \pi^{2}}{3}-x(2 \pi-x)\]on $[0,2 \pi)$ has Fourier cosine coefficients $4 \pi / k^{2}$ for $k \geq 1$. Hence or otherwise, investigate the
Suppose that $\eta \sim \operatorname{GP}(0, K)$, with $K$ as defined in the preceding exercise. The tied-down process $\zeta(t)=\eta(t)-\eta(0)$ is periodic and zero at integer multiples of
If you used the function $g l m(y \sim r x 2$, family=Gamma(link= identity)) in the preceding exercise, you may have experienced a failure to converge. Write your own Newton-Raphson function with
From the cosine integral $\int \cos (\omega t) e^{-|\omega|^{\alpha}} d \omega$, deduce that the $\alpha$-stable density has a Taylor series at the origin which begins\[\log p(t ; 1 / 2)=\text {
Check that the authors' lmer() code for the bee infection experiment produces the output shown in Sect. 9.2.2. Check that the revised lmer () code produces the output shown in Sect. 9.2.3.
The analysis of bee infection rates in Sect. 9.3 is only one of many similar analyses reported by Adler et al. (2020). A subsequent analysis examines how the mean infection intensity per colony is
All of the analyses of infection rates in Sect. 9.2 are for infections among surviving bumblebees, with the implicit assumption that the survival distribution does not depend on infection status.
Let $a$ be the vector of bird ages, and $n$ the vector of sequence lengths so that $\check{a}=n+1-a$ is reverse age. Show that $\operatorname{span}\{\mathbf{1}, a /(n+1)\}$ is equal to
Show that the model (10.1) implies exchangeability of terminal values $Y_{i, n_{i}} \sim$ $Y_{j, n_{j}}$ for every pair of birds, regardless of whether $n_{i}=n_{j}$ and regardless of the years
In the light of the preceding exercises, discuss the pros and cons of using normalized versus unnormalized age in the mean model (10.1).
Show that the extended model suggested in the last paragraph of Sect. 10.2 also has exchangeable initial values and exchangeable terminal values. Under what conditions on the parameter do initial
Discuss the connection between sampling consistency as described in Sect. 11.6.1 and lack of interference as described in Sect. 11.4.8.
According to the discussion of Dong et al. (2015) in Sect. 12.1, the real-estate market in Beijing is divided up into 1117 land parcels, which are partitioned into 111 districts. These administrative
A different version of the preceding model uses the complementary log-log link function:\[\log \left(-\log \left(1-\operatorname{pr}\left(Y_{i, j} \leq
BIC is a sub-model selection procedure. For an observation y∈Rny∈Rn, BIC adds the penalty −12dlogn−12dlog⁡n to the maximized loglog likelihood for sub-models of dimension dd. Deduce from
Let $Y_{1}, \ldots, Y_{n}$ be independent and identically distributed random variables whose distribution on $\mathbb{R}$ is atom-free, and let \(r_{i}=\operatorname{rank}\left(Y_{i}ight)
Let $Y_{1}, \ldots, Y_{n}$ be independent $N\left(\beta, \sigma^{2}ight)$ random variables with parameter $(\beta, \sigma)$ taking values in $\mathbb{R} \times \mathbb{R}^{+}$, and let the
For the pseudo log likelihood in the preceding exercise, show that the Fisher information matrix is diagonal and that it coincides with the Fisher information from the correct log likelihood.
Suppose that terminal values are conditionally independent given $\left(Y_{0}, Tight)$ with conditional distribution\[Y_{1, i} \sim N\left(\beta_{0}+\beta_{1} Y_{0, i}+\tau T_{i},
Let $\mathbf{t}$ be the treatment assignment vector, and let $B_{\mathbf{t}}$ be the associated block factor, i.e., $B_{\mathbf{t}}(i, j)=1$ if $t_{i}=t_{j}$ and zero otherwise. For \(g \in
Let $\mathbf{t}$ be the treatment assignment vector, and let $P_{W}$ be the $W$-orthogonal projection onto the subspace $\operatorname{span}(\mathbf{1}, \mathbf{t})$. Show that the
Under what conditions does the treatment model in the preceding exercise satisfy the lack of interference condition?
Show that the log likelihood function for the simple linear regression model is\[-n \log \sigma-\frac{1}{2} \sum\left(Y_{u}-\beta_{0}-\beta_{1} x_{u}ight)^{2} / \sigma^{2}\]Deduce that the triple
14.9 Show that the random-coefficient model (14.2) is equi-variant under affine covariate transformation $x \mapsto g_{0}+g_{1} x$ with $g_{1} eq 0$. Show that the induced transformation on
Suppose that the observations $Y_{1}, \ldots, Y_{n}$ in a two-arm randomized design are independent bivariate Gaussian with mean vector $\left(\mu_{1}, \mu_{2}ight)$ for units in the control arm,
A modification of the preceding model retains the mean vectors, while the covariance matrix is unrestricted but constant over units. Find an expression for the maximum-likelihood estimate of the
Show that the treatment effect in both preceding exercises is a group action on Gaussian distributions. What is the group, and how does it act? In only one case is the action on distributions induced
Let $\Theta_{0}=(0,1)$ and let $P_{\theta}$ be the iid Bernoulli model $\operatorname{Ber}(\theta)$. Let $\mathcal{G}$ be the additive group of addition modulo one, so that \(P_{g
By writing the complex vector $z$ and the Hermitian matrix $\Gamma$ as a linear combination of real and imaginary parts, show that the Hermitian quadratic form $z^{*} \Gamma z$ reduces to the
Show that the maximum-likelihood estimate of $\beta$ satisfies\[\left[X^{\prime} K^{\prime}\left(K V K^{\prime}ight)^{-1} K Xight] \hat{\beta}=X^{\prime} K^{\prime}\left(K V K^{\prime}ight)^{-1}
Deduce that the composite linear transformation $Y \mapsto L_{1} Y=Q \hat{\mu}$ is also a projection, and that it is the orthogonal projection whose image is the $p$-dimensional subspace \(Q
Show that the zero-mean exchangeable Gaussian process in Sect. 15.3.3 with covariances\[\operatorname{cov}\left(Y_{r}, Y_{s}ight)=\sigma_{0}^{2} \delta_{r s}+\sigma_{1}^{2}\]has a dynamic or
For fixed $v>0$, show that the Matérn measures are mutually consistent in the sense that $M_{d+1}(A \times \mathbb{R})=M_{d}(A)$ for all $d \geq 0$ and subsets $A \subset \mathbb{R}^{d}$. In
Consistency and finiteness together imply that the normalized Matérn measures define a real-valued process $X_{1}, X_{2}, \ldots$ in which $M_{n} / \Gamma(v)$ is the joint distribution of the
For the Matérn process, show that the sequence of partial averages $\bar{X}_{n}$ has a limit $\bar{X}_{\infty}=\lim _{n ightarrow \infty} \bar{X}_{n}$. For $n \geq 2$, what can you say about
One definition of the Bessel-K function is the integral\[\int_{0}^{\infty} \frac{\cos (\omega t) d \omega}{\left(1+\omega^{2}ight)^{v+1 / 2}}=\frac{\sqrt{ } \pi}{2^{v} \Gamma(v+1 / 2)} \times|t|^{v}
Special Gaussian family on $\mathbb{C}^{3}$ : Let $ho=\left(ho_{1}, ho_{2}, ho_{3}ight)$ be a real vector, and let $Z=\left(Z_{1}, Z_{2}, Z_{3}ight)$ be a zero-mean complex Gaussian variable
If $Z: \mathbb{R}^{3} ightarrow \mathbb{R}^{3}$ is the Gaussian process with parameter $(\omega, ho)$ as defined in the preceding exercise, and $R$ is a $3 \mathrm{D}$ rotation, show that the
Let $p, q$ be purely imaginary quaternions. Find the matrix representation $\chi_{4}(p \bar{q})$ of the quaternion product $p \bar{q}$, and show how this is related to $\chi(x)$ in (16.25)
Let $\psi_{0}(y)=e^{-y^{2} / 2} / \sqrt{2 \pi}$ be the standard normal density. Assume that $Y_{1}, \ldots, Y_{n}$ are independent standard normal. Show that the random variables $X_{i}=$
Show that the random variables $X_{i}=\psi_{1}\left(Y_{i}ight) / \psi_{0}\left(Y_{i}ight)$ in the preceding exercise have a density whose tail behaviour is $1 / f(x) \sim x^{2} \log (x)^{3 / 2}$
What does the preceding equation imply about the fraction of non-negligible signals among sites in the sample such that $\left|Y_{i}ight| \geq 3$ ?
Welham and Thompson (1997) discuss two possibilities for a Gaussian likelihood-ratio statistic. For arbitrary mean vector $\mu$, inner product matrix $W=$ $\Sigma^{-1}$, and $W$-orthogonal
For the balanced block design in the preceding exercise, show that the implied distribution for residuals is a two-parameter full exponential-family model with canonical sufficient statistic
Show that the REML estimate with positivity constraint satisfies $1+b \hat{\theta}=$ $\max (F, 1)$. What is the REML estimate for the second component? Express the constrained REML
To test for equality of variances in $k$ blocks of sizes $n_{1}, \ldots, n_{k}$, the REML procedure goes as follows. First, $\mathrm{bl} k$ is the $k$-dimensional subspace of
Let $Y$ be a non-negative random variable with cumulants $\kappa_{r}$ such that $\kappa_{r} / \kappa_{1}^{r}=$ $O\left(ho^{r-1}ight)$ as $ho ightarrow 0$. In other words, the scale-free
Show that the transformation $\mathbb{R}^{n} ightarrow \mathbb{R}^{n}$ defined by\[\bar{u} \mapsto \bar{u}+\text { const }, \quad u_{i}-\bar{u} \mapsto \lambda\left(u_{i}-\bar{u}ight)\]is linear
As a function of $\lambda$, show that the transformation $\mathbb{R}_{+}^{n} ightarrow \mathbb{R}^{n}$\[(g y)_{i}=\frac{y_{i}^{\lambda}-1}{\lambda \dot{y}^{\lambda-1}}\]is continuous at

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