Bainbridge's mass spectrometer, separates ions having the same velocity. The ions, after entering through slits $S_1$ and $S_2$, pass through a velocity selector composed of an electric field produced by the charged plates P and $\mathbf{P}^{\prime}$, and a magnetic field $\overrightarrow{\mathbf{B}}$ perpendicular to the electric field and the ion path. Those ions that pass undeviated through the crossed $\overrightarrow{\mathbf{E}}$ and $\overrightarrow{\mathbf{B}}$ fields enter into a region where a second magnetic field $\vec{B}^{\prime}$ exists, and are bent into circular paths. A photographic plate registers their arrival. Show thal $q / m=E / r B B^{\prime}$, where $r$ is the radius of the circular orbit.

What are examples of plant diseases that water molds cause?

The farmer's wife collects snakes and iguanas. One day she observed that her reptiles, which are normal, have a total of $60$ eyes and $68$ feet. How many reptiles of each type does she have?

Give two examples of sustainable development.

A proton traveling at 23.0$^\circ$ with respect to the direction of a magnetic field of strength 2.60 mT experiences a magnetic force of $6.50 \times 10^{-17} N$. Calculate its kinetic energy in electron volts.

A proton traveling at 23.0$^\circ$ with respect to the direction of a magnetic field of strength 2.60 mT experiences a magnetic force of $6.50 \times 10^{-17} N$. Calculate the proton’s speed.

Atom 1 of mass 35 u and atom 2 of mass 37 u are both singly ionized with a charge of +e. After being introduced into a mass spectrometer and accelerated from rest through a potential difference V = 7.3 kV, each ion follows a circular path in a uniform magnetic field of magnitude B = 0.50 T. What is the distance Δx between the points where the ions strike the detector?

Let X be a Poisson random variable with parameter

$$\lambda$$

. What value of λ maximizes P{X=k},

$$k \geq 0 ?$$

Suppose that X is a Poisson random variable with parameter λ. Let the prior distribution for λ be a gamma distribution with parameters m + 1 and

$$(m + 1)/λ_0$$

. a. Find the posterior distribution for λ. b. Find the Bayes estimator for λ.

Let $Z \sim \mathcal{N}(0, 1)$, and $c$ be a nonnegative constant. Find $E(\max(Z − c, 0))$, in terms of the standard Normal CDF $\Phi$ and PDF $\varphi$. (This kind of calculation often comes up in quantitative finance.) Hint: Use LOTUS, and handle the max symbol by adjusting the limits of integration appropriately. As a check, make sure that your answer reduces to $\frac{1}{\sqrt{2\pi}}$ when $c = 0$.

The Gumbel distribution is the distribution of $− \log X$ with $X \sim \text{Expo}(1)$. (a) Find the CDF of the Gumbel distribution. (b) Let $X_1,X_2, . . .$ be i.i.d. $\text{Expo}(1)$ and let $M_n = \max (X_1, . . . , X_n)$. Show that $M_n − \log n$ converges in distribution to the Gumbel distribution, i.e., as $n \rightarrow \infty$ the CDF of $M_n − \log n$ converges to the Gumbel CDF.

The Laplace distribution has PDF

$$f ( x ) = \frac { 1 } { 2 } e ^ { - | x | }$$

for all real $x$. The Laplace distribution is also called a symmetrized Exponential distribution. Explain this in the following two ways. (a) Plot the PDFs and explain how they relate. (b) Let $X \sim \text{Expo}(1)$ and $S$ be a random sign (1 or −1, with equal probabilities), with $S$ and $X$ independent. Find the PDF of $SX$ (by first finding the CDF), and compare the PDF of $SX$ and the Laplace PDF.

Let $X$ and $Y$ be i.i.d. $\text{Expo}(1)$, and $L = X − Y$ . The Laplace distribution has PDF

$$f ( x ) = \frac { 1 } { 2 } e ^ { - | x | }$$

for all real $x$. Use MGFs to show that the distribution of $L$ is Laplace.

Let $U_1, . . . , U_n$ be i.i.d. $\text{Unif}(0, 1)$, and $X = \max(U_1, . . . , U_n)$. What is the PDF of $X$? What is ,$EX$? Hint: Find the CDF of $X$ first, by translating the event $X \leq x$ into an event involving $U_1, . . . , U_n$.