A-level Further Mathematics₇₃₆₇

Understand expectation of linear functions of DRVs and know the formulae:

$\mathsf{E}\left(\mathit{a}\mathit{X}\mathsf{+}\mathit{b}\right)\mathsf{=}\mathit{a}\mathsf{E}\left(\mathit{X}\right)\mathsf{+}\mathit{b}\mathit{}$ and $\mathsf{V}\mathsf{a}\mathsf{r}\mathit{}\left(\mathit{a}\mathit{X}\mathsf{+}\mathit{b}\right)\mathsf{=}\mathit{}{\mathit{a}}^{\mathsf{2}}\mathsf{V}\mathsf{a}\mathsf{r}\mathit{}\mathsf{\left(}\mathit{X}\mathsf{\right)}$

Know the formula $\mathsf{E}\left(\mathrm{g}\left(\mathit{X}\right)\right)\mathsf{=}\mathit{}\mathsf{\sum }\mathrm{g}\mathsf{\left(}{\mathit{x}}_{\mathit{i}}\mathsf{\right)}{\mathit{p}}_{\mathit{i}}$

Find the mean, variance and standard deviation for functions of a DRV such as $\mathsf{E}\left(\mathsf{5}{\mathit{X}}^{\mathsf{3}}\right)\mathsf{,}\mathit{}\mathit{}\mathsf{E}\left(\mathsf{18}{\mathit{X}}^{\mathsf{-}\mathsf{3}}\right)\mathsf{,}\mathit{}\mathsf{V}\mathsf{a}\mathsf{r}\mathit{}\mathsf{\left(}\mathsf{6}{\mathit{X}}^{\mathsf{-}\mathsf{1}}\mathsf{\right)}$

Know the discrete uniform distribution defined on the set $\left\{\mathsf{1,2}\mathsf{,}\mathsf{\dots }\mathsf{,}\mathit{n}\right\}$ . Understand when this distribution can be used as a model.

Understand conditions for a Poisson distribution to model a situation. Understand terminology $\mathit{X}\mathit{}\mathsf{~}\mathit{}\mathsf{P}\mathsf{o}\mathsf{\left(}\mathit{\lambda }\mathsf{\right)}$ .

Know mean, variance and standard deviation of a Poisson distribution.

Use the result that, if $\mathit{X}\mathit{}\mathsf{~}\mathit{}\mathsf{P}\mathsf{o}\left(\mathit{\lambda }\right)$ then the mean and variance of $\mathit{X}$ are equal.

Formulate hypotheses and carry out a hypothesis test of a population mean from a single observation from a Poisson distribution using direct evaluation of Poisson probabilities.

Understand Type $\mathrm{I}$ and Type $\mathrm{II}$ errors and define in context. Calculate the probability of making a Type $\mathrm{I}$ error from tests based on a Poisson or Binomial distribution.

Calculate probability of making Type $\mathrm{I}$ error from tests based on a normal distribution.

Understand the power of a test. Calculations of P(Type $\mathrm{II}$ error) and power for a test for tests based on a normal, Binomial or a Poisson distribution.

Find the mean, variance and standard deviation for a given pdf, $\mathrm{f}\mathsf{\left(}\mathit{x}\mathsf{\right)}$ . Know the formulae

$\mathsf{E}\left(\mathit{X}\right)\mathsf{=}\mathit{}\mathsf{\int }\mathit{x}\mathrm{f}\left(\mathit{x}\right)\mathrm{d}\mathit{x}$ , $\mathsf{E}\left({\mathit{X}}^{\mathsf{2}}\right)\mathsf{=}\mathit{}\mathsf{\int }{\mathit{x}}^{\mathsf{2}}\mathrm{f}\left(\mathit{x}\right)\mathrm{d}\mathit{x}$ , $\mathsf{V}\mathsf{a}\mathsf{r}\left(\mathit{X}\right)\mathsf{=}\mathsf{E}\left({\mathit{X}}^{\mathsf{2}}\right)\mathsf{-}\mathsf{(}\mathsf{E}\left(\mathit{X}\right){\mathsf{)}}^{\mathsf{2}}$

Understand the expectation and variance of linear functions of CRVs and know the formulae:

$\mathsf{E}\left(\mathit{a}\mathit{X}\mathsf{+}\mathit{b}\right)\mathsf{=}\mathit{a}\mathsf{E}\left(\mathit{X}\right)\mathsf{+}\mathit{b}$ and $\mathsf{V}\mathsf{a}\mathsf{r}\mathit{}\left(\mathit{a}\mathit{X}\mathsf{+}\mathit{b}\right)\mathsf{=}\mathit{}{\mathit{a}}^{\mathsf{2}}\mathsf{V}\mathsf{a}\mathsf{r}\mathit{}\mathsf{\left(}\mathit{X}\mathsf{\right)}$

Know the formula $\mathsf{E}\left(\mathrm{g}\left(\mathit{X}\right)\right)\mathsf{=}\mathit{}\mathsf{\int }\mathrm{g}\left(\mathit{x}\right)\mathrm{f}\mathsf{\left(}\mathit{x}\mathsf{\right)}\mathrm{d}\mathit{x}$

Find the mean, variance and standard deviation of functions of a continuous random variable such as

$\mathsf{E}\left(\mathsf{5}{\mathit{X}}^{\mathsf{3}}\right)\mathsf{,}\mathit{}\mathit{}\mathsf{E}\left(\mathsf{18}{\mathit{X}}^{\mathsf{-}\mathsf{3}}\right)\mathsf{,}\mathit{}\mathsf{V}\mathsf{a}\mathsf{r}\mathit{}\mathsf{\left(}\mathsf{6}{\mathit{X}}^{\mathsf{-}\mathsf{1}}\mathsf{\right)}$

Understand and use a cumulative distribution function, $\mathrm{F}\mathsf{\left(}\mathit{x}\mathsf{\right)}$ . Know the relationship between $\mathrm{f}\mathsf{\left(}\mathit{x}\mathsf{\right)}$ and $\mathrm{F}\mathsf{\left(}\mathit{x}\mathsf{\right)}$ .

$\mathrm{F}\left(\mathit{x}\right)\mathsf{=}\mathit{}{\mathsf{\int }}_{\mathsf{-}\mathit{\infty }}^{\mathit{x}}\mathrm{f}\left(\mathit{t}\right)\mathrm{d}\mathit{t}\mathit{}$ and $\mathrm{f}\left(\mathit{x}\right)\mathsf{=}\frac{\mathrm{d}}{\mathrm{d}\mathit{x}}\mathrm{F}\left(\mathit{x}\right)$

Know that if $\mathit{X}$ and $\mathit{Y}$ are independent (discrete or continuous) random variables then $\mathsf{E}\left(\mathit{X}\mathsf{+}\mathit{Y}\right)\mathsf{=}\mathsf{E}\left(\mathit{X}\right)\mathsf{+}\mathsf{E}\left(\mathit{Y}\right)$ and $\mathsf{V}\mathsf{a}\mathsf{r}\left(\mathit{X}\mathsf{+}\mathit{Y}\right)\mathsf{=}\mathsf{V}\mathsf{a}\mathsf{r}\left(\mathit{X}\right)\mathsf{+}\mathsf{V}\mathsf{a}\mathsf{r}\mathsf{\left(}\mathit{Y}\mathsf{\right)}$

Use of $\mathsf{\sum }\frac{{\mathsf{(}{\mathit{O}}_{\mathit{i}}\mathsf{-}{\mathit{E}}_{\mathit{i}}\mathsf{)}}^{\mathsf{2}}}{{\mathit{E}}_{\mathit{i}}}$ as an approximate ${\text{χ}}^{\mathsf{2}}$ statistic with appropriate degrees of freedom.

Test for the mean of a normal distribution with unknown variance using a

$\mathit{t}$ with appropriate degrees of freedom.

Construct symmetric confidence intervals for the mean of a normal distribution with known variance.