$[$Please answer all questions below in ghe figure, make sure uou select an answer from the given ones, will thumb up$.$ Thanks

Prompt $1]$ For Questions $1-5,$ consider the p$-$dimensional hypercube $H_1^p=[-1,1{]}^p.$ Assume we draw two $p-$dimensional points $a=(a_1,a_2,$dots,$a_p)$ and $b=(b_1,b_2,$dots,$b_P)$ uniformly at random from this hypercube.

Denote by $H_r^p$ the p$-$dimensional hypercube $[-r,r{]}^p$ for What is the probability of the event $a!in{H}_r^p?$

Hint: Notice that, if you choose a point uniformly at random from a set $Sin{R}^p,$ the probability that this point is in another set TsubS is equal to the ratio of the volumes, i$.$e$.,\frac{\text{V}\text{o}\text{l}(T)}{\text{V}\text{o}\text{l}(S)}.$

$r^p$

$r^p$

$1-r^p$

$r^{\{p-1\}}$

$[$Refer to Prompt $1]$ What is the value of $r$ for which $P(a!in{H}_r^p)=90\%$ as a function of $p?$

$r={10}^{\{-\frac{1}{p}\}}$

$r={90}^{\{-\frac{1}{p}\}}$

$r=0\mathrm{.}1$

$r=0\mathrm{.}9$

$[$Refer to Prompt $1]$ Using the expression for $r$ obtained in Question $2,$ what is the limit of $r$ when $p?$ Please, think about the interpretation of this result.

$0$

$1$

$0\mathrm{.}5$

$\frac{1}{p}$

$[$Refer to Prompt $1]$ Compute the expected squared distance between the points a and $b,$ i$.$e$.E[||a-b|{|}^2].$

Hint: Remember that $||a-b|{|}^2={\displaystyle \underset{i=1}{\overset{p}{}}}(a_i-b_i{)}^2.$ Also, notice that, since a and $b$ are drawn uniformly at random from a hypercube, the coordinates $a_i$ and $b_i$ are independently drawn uniformly at random from the interval $-1,1$; hence, their mean and variance are, respectively, $0$ and $\frac{1}{3}.$

$\frac{p}{3}$

$\frac{p}{2}$

$\frac{2p}{3}$

$p$

$[$Refer to Prompt $1]$ Compute the conditional expectation $|]=[(0,0,$dots,$0)$

$\frac{p}{3}$

$\frac{p}{2}$

$\frac{2p}{3}$

$p$