Examples of beta in the following topics:
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- Beta decay is a type of radioactive decay in which a beta particle (an electron or a positron) is emitted from an atomic nucleus.
- There are two types of beta decay.
- Beta minus (β) leads to an electron emission (e−); beta plus (β+) leads to a positron emission (e+).
- Beta decay is mediated by the weak force.
- The continuous energy spectra of beta particles occur because Q is shared between a beta particle and a neutrino.
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- $\displaystyle \beta \equiv \frac{\bf u}{c},~\text{so}~ \kappa = 1 - {\bf n} \cdot \beta$
- $\displaystyle {\bf E}(r,t) = \kern-2mm q \left [ \frac{({\bf n} - \beta)(1-\beta^2)}{\kappa^3 R^2} \right ]_\mathrm{ret}\!
- \frac{q}{c} \left [ \frac{\bf n}{\kappa^3 R} \times \left [ ({\bf n} - \beta ) \times \dot{\beta} \right ]\right ]_\mathrm{ret} \\ {\bf B}(r,t) = \kern-2mm\left [ {\bf n} \times {\bf E}(r,t) \right ]_\mathrm{ret}.$
- $\displaystyle {\bf E}_{rad}(r,t) = + \frac{q}{c} \left [ \frac{\bf n}{\kappa^3 R} \times \left [ ({\bf n} - \beta ) \times \dot{\beta} \right ]\right ] \\ {\bf B}_{rad}(r,t) = \left [ {\bf n} \times {\bf E}_{rad}(r,t) \right ].$
- $\displaystyle {\bf S} = {\bf n} \frac{q^2}{4\pi c \kappa^6 R^2} \left | {\bf n} \times \left \{ \left ( {\bf n} - \beta \right ) \times {\dot{\beta}} \right \} \right |^2$
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- $\begin{aligned}
\cos(\alpha + \beta) &= \cos \alpha \cos \beta - \sin \alpha \sin \beta \\
\cos(\alpha - \beta) &= \cos \alpha \cos \beta + \sin \alpha \sin \beta
\end{aligned}$
- $\begin{aligned}
\sin(\alpha + \beta) &= \sin \alpha \cos \beta + \cos \alpha \sin \beta \\
\sin(\alpha - \beta) &= \sin \alpha \cos \beta - \cos \alpha \sin \beta
\end{aligned}$
- $\displaystyle{
\begin{aligned}
\tan(\alpha + \beta) &= \frac{ \tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta} \\
\tan(\alpha - \beta) &= \frac{ \tan \alpha - \tan \beta}{1 + \tan \alpha \tan \beta}
\end{aligned}
}$
- Apply the formula $\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta$:
- We can thus apply the formula for sine of the difference of two angles: $\sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta$.
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- $\displaystyle R {\hat E} (\omega) = \frac{q}{2\pi c} \int_{-\infty}^{\infty} \left [ \frac{\bf n}{\kappa^3} \times \left [ ({\bf n} - \beta ) \times \dot{\beta} \right ]\right ]_\mathrm{ret} e^{i\omega t} d t.$
- $\displaystyle R {\hat E} (\omega) = \frac{q}{2\pi c} \int_{-\infty}^{\infty} \left [ \frac{\bf n}{\kappa^2} \times \left [ ({\bf n} - \beta ) \times \dot{\beta} \right ]\right ] e^{i\omega (t'+R(t')/c)} d t'.$
- $\displaystyle R {\hat E} (\omega) = \frac{q}{2\pi c} \int_{-\infty}^{\infty} \left [ \frac{\bf n}{\kappa^2} \times \left [ ({\bf n} - \beta) \times \dot{\beta} \right ]\right ] e^{i\omega (t'-{\bf n}\cdot {\bf r}(t')/c)} d t'.$
- $\displaystyle \frac{d W}{d\Omega d\omega} = \frac{q^2}{4\pi^2 c} \left | \int_{-\infty}^{\infty} \frac{{\bf n} \times \left [ ({\bf n} - \beta ) \times \dot{\beta} \right ]}{\left ( 1-\beta\cdot {\bf n} \right )^2} e^{i\omega (t'-{\bf n}\cdot {\bf r}(t')/c)} d t' \right |.$
- $\displaystyle \frac{{\bf n} \times \left [ ({\bf n} - \beta ) \times \dot{\beta} \right ]}{\left ( 1-\beta\cdot {\bf n} \right )^2} = \frac{d}{d t'} \left [ \frac{{\bf n} \times ({\bf n} \times \beta ) }{1-\beta\cdot {\bf n}} \right ].$
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- A portfolio's Beta is the volatility correlated to an underlying index.
- A portfolio's Beta is the volatility correlated to an underlying index.
- What would the following portfolios have for Beta values?
- Thus, the portfolio would have a Beta value of 3.
- Two hypothetical portfolios; what do you think each Beta value is?
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- $\displaystyle I'(E',\mu') = F_0 \left (\frac{E'}{E}\right)^2 \delta (E-E_0)\\ \displaystyle = F_0 \left (\frac{E'}{E_0}\right)^2 \delta (\gamma E' (1+\beta\mu') -E_0) \\ \displaystyle = \frac{F_0}{\gamma\beta E'} \left (\frac{E'}{E_0}\right)^2 \delta \left (\mu' - \frac{E_0-\gamma E'}{\gamma\beta E'} \right )$
- $\displaystyle j'(E_f') = \frac{N' \sigma_T E_f' F_0}{2 E_0^2 \gamma \beta}~\text{ if }~ \frac{E_0}{\gamma (1+\beta)} < E_f' < \frac{E_0}{\gamma(1-\beta)}$
- $\displaystyle j(E_f,\mu_f) = \frac{E_f}{E_f'} j'(E_f') \\ \displaystyle = \frac{N \sigma_T E_f F_0}{2 E_0^2 \gamma^2 \beta} \\ \displaystyle ~~\text{ if }~ \frac{E_0}{\gamma (1+\beta)(1-\beta \mu_f)} < E_f < \frac{E_0}{\gamma(1-\beta)(1-\beta \mu_f)} \nonumber$
- $\displaystyle \frac{1}{\beta} \left [ 1 - \frac{E_0}{E_f} \left ( 1 + \beta \right ) \right ] < \mu_f < \frac{1}{\beta} \left [ 1 - \frac{E_0}{E_f} \left ( 1 - \beta \right ) \right ].$
- $\displaystyle j(E_f) = \frac{N \sigma_T F_0}{4 E_0 \gamma^2 \beta^2} \left \{ \begin{array}{lc} (1+\beta) \frac{E_f}{E_0} - (1 -\beta ), \frac{1-\beta}{1+\beta} < \frac{E_f}{E_0} < 1 \\ \displaystyle (1+\beta) - \frac{E_f}{E_0} (1 -\beta ), 1 < \frac{E_f}{E_0} < \frac{1+\beta}{1-\beta} \\ \displaystyle 0, \text{ otherwise } \end{array} \right .$
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- Systematic risk can be understood further using the measure of Beta.
- Betas less than 0: Asset generally moves in the opposite direction as compared to the index.
- Betas equal to 0: Movement of the asset is uncorrelated with the movement of the benchmark.
- Beta is a measure that relates the rate of return of an asset, ra, with the rate of return of a benchmark, rb.
- Use a stock's beta to estimate a stock's daily growth or decline.
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- ${ Y }_{ i }=\alpha +\beta { x }_{ i }+{ \varepsilon }_{ i }$
- It is desired to test the null hypothesis that the slope $\beta$ is equal to some specified value $\beta_0$ (often taken to be $0$, in which case the hypothesis is that $x$ and $y$ are unrelated).
- Let $\hat{\alpha}$ and $\hat{\beta}$ be least-squares estimators, and let $SE_\hat{\alpha}$ and $SE_\hat{\beta}$, respectively, be the standard errors of those least-squares estimators.
- $\displaystyle{SE_{\hat{\beta}}=\frac{\sqrt{\frac{1}{n-2} \sum_{i=1}^n \left(Y_i - \hat{y}_i \right) ^2}}{\sqrt{\sum_{i=1}^n \left( x_i - \bar{x} \right) ^2}}}$
- $\displaystyle{t = \frac{ \left( \hat{\beta} - \beta_0\right) \sqrt{n-2}}{\sqrt{ \frac{\text{SSR}}{\sum_{i=1}^n\left( x_i - \bar{x}\right)^2}}}}$
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- $\displaystyle {\bf S} = {\bf n} \frac{q^2}{4\pi c \kappa^6 R^2} \left | {\bf n} \times \left \{ \left ( {\bf n} - \beta\right ) \times {\dot{\beta}} \right \} \right |^2$
- $\displaystyle \frac{dP(t')}{d\Omega} = \frac{q^2}{4\pi c} \frac{ \left | {\bf n} \times \left \{ \left ( {\bf n} - \beta \right ) \times {\dot{\beta}} \right \} \right |^2}{\left ( 1 - {\bf n} \cdot \beta \right )^5} $
- Let's start by assuming the $\beta$ is parallel to ${\dot{\beta}}$, so $\beta \times {\dot{\beta}}=0$.
- Let's repeat the calculation for circular motion, in which $\beta \perp {\dot{\beta}}$.
- $\displaystyle \frac{dP(t')}{d\Omega} = \frac{q^2 {\dot u}^2}{4\pi c^3} \frac{ 1 }{\left ( 1 - \beta \cos\Theta \right )^3} \left [ 1 - \frac{ \sin^2 \Theta \cos^2\phi }{\gamma^2 \left ( 1 - \beta \cos\Theta \right )^2} \right ]$
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- When $a$ is equal to one, $\alpha_1$ and $\alpha_2$ both equal one, and $\beta_1$ and $\beta_2$ are factors of the constant $c$ such that:
- Such that $b = \alpha_1 \beta_2 + \alpha_2 \beta_1$.