Uranium-235 undergoes a series of a-particle and $\beta$-particle productions to end up as lead-207. How many $\alpha$ particles and $\beta$ particles are produced in the complete decay series?

All else being equal, is a put option on a high-beta stock worth more than one on a low-beta stock? The firms have identical firm-specific risk.

Which of the following functions f, defined on vectors $\alpha=\left(x_{1}, x_{2}\right) \text { and } \beta=\left(y_{1}, y_{2}\right)$ in $R^2$, are bilinear forms? (a) $f(\alpha, \beta)=1$. (b) $f(\alpha, \beta)=\left(x_{1}-y_{1}\right)^{2}+x_{2} y_{2}$. (c) $f(\alpha, \beta)=\left(x_{1}+y_{1}\right)^{2}-\left(x_{1}-y_{1}\right)^{2}$. (d) $f(\alpha, \beta)=x_{1} y_{2}-x_{2} y_{1}$.

Which of the following characteristic $\mathrm{x}$ rays has the longest wavelength? (a) $\mathrm{K}_\alpha ;$ (b) $\mathrm{K}_\beta ;$ (c) $\mathrm{L}_\alpha$; (d) $\mathrm{L}_\beta$.

An $\beta$ Oxidation: How Many Cycles? How many cycles of $\beta$ oxidation are required for the complete oxidation of activated oleic acid, $18: 1\left(\Delta^9\right)$ ?

Use a determinant to show that the matrix

$$\left[\begin{array}{ccc}{\sin ^{2} \alpha} & {\sin ^{2} \beta} & {\sin ^{2} \gamma} \\ {\cos ^{2} \alpha} & {\cos ^{2} \beta} & {\cos ^{2} \gamma} \\ {1} & {1} & {1}\end{array}\right]$$

is not invertible for any values of $\alpha, \beta$, $\gamma$.

Consider $F(\omega)=e^{-\beta \omega}, \beta>0(\omega \geq 0).$ (a) Derive the inverse Fourier sine transform of $F(\omega).$ (b) Derive the inverse Fourier cosine transform of $F(\omega).$

Let

$$Y _ { 1 } , Y _ { 2 } , \ldots , Y _ { n }$$

denote a random sample from an exponentially distributed population with density

$$f ( y | \theta ) = \theta e ^ { - \theta y } , 0 < y$$

. (Note: the mean of this population is

$$\mu = 1 / \theta$$

.) Use the conjugate gamma

$$( \alpha , \beta )$$

prior for

$$\theta$$

to do the following. Show that the posterior density for

$$\theta | \left( y _ { 1 } , y _ { 2 } , \ldots , y _ { n } \right)$$

is a gamma density with parameters

$$\alpha ^ { * } = n + \alpha \text { and } \beta ^ { * } = \beta / \left( \beta \sum y _ { i } + 1 \right)$$

.

D-Arabinose can exist in both pyranose and furanose forms. a. Draw the

$$\alpha \text { and } \beta$$

anomers of D-arabinofuranose. b. Draw the

$$\alpha \text { and } \beta$$

anomers of D-arabinopyranose.

The probability of a system being in the ith microstate is $P_{i}=\mathrm{e}^{-\beta E_{i}} / Z$, where $E_{i}$ is the energy of the ith microstate and $\beta \text { and } Z$ are constants. Show that the entropy is given by $S / k_{\mathrm{B}}=\ln Z+\beta U$, where $U=\sum_{i} P_{i} E_{i}$ is the internal energy.

Let $\alpha, \beta, \gamma, \delta$ be real numbers. We ask when it is possible to find a polynomial f over R, of degree not more than 2, such that $f(-1)=\alpha, f(1)=\beta, f(3)=\gamma$ and $f(0)=\delta$. Prove that this is possible if and only if $3 \alpha+6 \beta-\gamma-8 \delta=0$.

Let X and P be random variables. Suppose that the conditional distribution of X given P=p is the binomial distribution with parameters $n$ and $p$. Suppose that the distribution of $P$ is the beta distribution with parameters $\alpha=1$ and $\beta=1$. Find the marginal distribution of $X$.

Choose the correct letter that best matches the phrase. a. amplitude b. midline c. period d. vertical shift e. double-angle formula f. half-angle formula g. difference of angles formula h. phase shift

$$\cos (\alpha-\beta)=\cos \alpha \cos \beta+\sin \alpha \sin \beta$$

Bi-$210$ is a beta emitter with a half-life of $5.0$ days. If a sample contains $1.2 \mathrm{~g}$ of Bi-$210$, how many beta emissions occur in $5.0$ days?