# Problem 1

## Problem (i)

\begin{align*} F(x) &= P(X \leq x\\ &= \sum_{A} P(X \leq x, A)\\ &= P(X \leq x, A) + P(X \leq x, A^c)\\ &= P(X \leq x|A)P(A) + P(X \leq x|A^c)P(A^c)\\ &= F_1(x)\theta + F_2(x)(1-\theta) \end{align*}

To deduce the next equation, we simply differentiate the above:

\begin{align*} \frac{d}{dx} F(x) &= \frac{d}{dx} F_1(x)\theta+F_2(x)(1-\theta)\\ f(x) &= f_1(x)\theta+f_2(x)(1-\theta)\\ \int x f(x)dx &= \int x\theta f_1(x)dx + \int x(1-\theta)f_2(x)dx\\ E[X] &= \theta \mu_1 + (1-\theta)\mu_2 \end{align*}

## Problem (ii)

\begin{align*} W_i &\sim \mathcal{N}(4.5,0.25)\\ S_6=\sum_{i=1}^6 W_I &\sim \mathcal{N}(27,1.5)\\ P(S_6\leq 25) &= P(\frac{S_6-27}{\sqrt{1.5}} \leq -\frac{2}{\sqrt{1.5}})\\ &= P(Z \leq -1.63)\\ &=0.056 \end{align*}

## Problem (iii)

\begin{align*} EW &= P(S_6 \leq 25)E[S_7] + P[S_6 \geq 25]E[S_6]\\ &= 0.056*31.5+(1-0.056)*27\\ &= 27.322 \end{align*}

## Problem (iv)

\begin{align*} P(S_6 \leq 25) &= 0.01\\ P(Z \leq -\frac{2}{\sigma}) &= 0.01 \implies -\frac{2}{\sigma}&= -2.33\\ &= 1.165 \end{align*}

# Problem 2

## Problem (i)

Median = $$\int_{-\infty}^m = \frac{1}{2}$$ Thus,

\begin{align*} \int_{-\infty}^m xf(x) dx &= \frac{1}{2}\\ \int_{-p}^{\infty} x f(-x) = \frac{1}{2}\\ &= \int_p^{\infty} xf(x)\\ &= \int_{-p}^{\infty} xf(-x)dx\\ \implies p &=0 \end{align*}

Thus median $$m=0$$

## Problem (ii)

\begin{align*} EX &= \int_{-\infty}^{\infty}xf(x)dx \\ &= \int_{\infty}^{-\infty}-xf(-x)d(-x) \ x \longrightarrow -x\\ &= -\int_{-\infty}^{\infty}xf(-x)dx\\ &= -\int_{-\infty}^{\infty}xf(x)dx \text{ since } f(x)=f(-x) \\ \implies E[X] &=0 \end{align*}

## Problem (iii)

$$Y=X^2$$

\begin{align*} E[XY] &= E[X^3]\\ &= \int_{-\infty}^\infty x^3f(x)dx\\ &=0 \end{align*}

$$Cov(X<y)=E[XY]-E[X]E[Y] = 0$$ and hence $$X,Y$$ are uncorrelated

## Problem (iv)

\begin{align*} g(y) &= f_X(\sqrt{y}) \frac{dx}{dy}\\ &= \frac{1}{2\sqrt{y}}f_X(\sqrt{y})\\ \text{Note the typo in the original question where $\frac{1}{2}$ is issing} \end{align*}

# Problem 3

## Problem 3.(i)

\begin{align*} f(x,y) &= \frac{1}{x}e^{-x} \\ f(x) &= \int_0^{x} f(x,y)dy\\ &= e^{-x}\\ f_{Y|X}(x)&=\frac{f(X=x,Y)}{f(X)}\\ &= \frac{1}{x} \end{align*}

## Problem 3.(ii)

\begin{align*} E[X^mY^n] &= \int_{y}^ix^my^n\frac{1}{x}e^{-x} dxdyi\\ &= \int_0^\infty x^{m-1e^{-xdx} \int_0^x y^mdy}\\ &= \int_0^\infty x^{m-1+n+1}e^{-x}\frac{1}{n}\\ &= \frac{(m+n)!}{n+1} \end{align*}

## Problem 3.(iii)

\begin{align*} E[X]&=1\\ E[X^2] &= 2\\ Var(X) &= 1\\ E[Y]&= 1/2\\ E[Y^2] &= 2/3\\ Var(Y) &= 5/12 \end{align*}

# Problem 4

## Problem 4.(i)

\begin{array}{|c|c|c|} X1/X2 & 0 & 1\\\hline 0 & 0.1 & 0.1\\\hline 1 & 0.3 & 0.5\\\hline \end{array}

## Problem 4.(ii)

\begin{align*} S_Y &= \{0, 1, 2, 3\}\\ \end{align*}

Idea: Start of with a random row and column in the table, Then keep sampling until there are 10 samples(Ignoring anything that is not in $$S_Y$$)

## Problem 4.(iii)

This is straightforward based on theses caseSs

\begin{align*} Y&=0 \implies X_1=0,X_2=0\\ Y&=1 \implies X_1=0,X_2=1\\ Y&=2 \implies X_1=1, X_2=0\\ Y&=3 \implies X_1=1,X_2=1 \end{align*}

TODO

# Problem 5

## Problem 5.(i)

\begin{align*} cor(X_1,X_2) &= \frac{Cov(X_1,X_2)}{\sigma_1\sigma_2}\\ &= \frac{2}{4}\\ &= 1/2 \end{align*}

## Problem 5.(ii)

\begin{align*} E[Y] &= -1+1 =0\\ Var(Y) &= Var(X_1) + Var(X_2) + 2Cov(X_1,X_2)\\ &= 21 \end{align*}

## Problem 5.(ii)

Since $$Y_1.Y_2$$ are both normal, independnce requires just the covariance being zero.

\begin{align*} Cov(Y_1,Y_2) &=0\\ &= KVar(X_1)+ (k+1)Cov(X_1,X_2) + Var(X_2)\\ &= 3k+18\\ \implies k &=-6 \end{align*}

## Problem 5.(b)

$$E[\mathbf{Y}]=\mathbf{0}$$

$$V= \begin{pmatrix} 1/3 & 2/9 & 1/9 & 0 & \dots & 0\\ 2/9 & 2/9 & 1/9 & 0 & \dots & 0\\ 1/9 & 1/9 & 1/9 & 0 & \dots & 0\\ 0 & 0 & 0 & 1/3 & 2/9 & 1/9\\ \end{pmatrix}$$ Essentiall th 3 rows block shifts every 4th row.

Joint distribution of $$Y_1,Y_2,\dots, Y_{n-2}$$ follows a MVN with mean 0 and variance as the above matrix

# Problem 6

\begin{align*} M_X(t) &= E[e^{tX}]\\ &= \int_0^\infty \frac{e^{tx} \theta^\alpha x^{\alpha-1} e^{-\theta x}}{\Gamma(\alpha)}dx\\ &= \int_0^\infty \frac{(\theta-t)^\alpha}{(\theta-t)^\alpha} \frac{ \theta^\alpha x^{\alpha-1} e^{t-\theta x}}{\Gamma(\alpha)}\\ &= \frac{\theta^\alpha}{(t-\theta)^\alpha}\\ EX &= M_X'(0)\\ &= \alpha\theta^\alpha(\theta-t)^{-\alpha-1}\\ &= \alpha/\theta\\ EX^2 &= M_X''(0)\\ &= \alpha(\alpha-1)/\theta^2\\ Var(X) = \alpha/\theta^2-\alpha(\alpha-1)/\theta^2\\ &= \alpha/\theta^2 \end{align*}

## Problem 6.(ii)

\begin{align*} M_X(t) &= \frac{\theta}{\theta-t} Z &= \sum_iX_i\\ M_Z(t) &= \prod M_{X_i}(t)\\ &= (\frac{\theta}{\theta-t})^n\\ &\sim \Gamma(\theta,n) \end{align*}

## Problem 6.(iii)

Central Limit Theorem:

If $$X_i$$ represent random variables whose mgf exits in a neighborhood of 0 and has finite first and second momements, then $$\frac{\bar{X_n}-\mu}{\sigma/\sqrt{n}} \sim \mathcal{N}(0,1)$$

$$\Gamma(n,\theta) \sim \mathcal{N}(n\sqrt{n}/\theta,n^2/\theta^2 )$$

# Problem 7

## Problem 7(a)

\begin{align*} P[X_1=1] &= \frac{m}{n} \\ P[X_2=1] &= \sum_{{x_1}} P(X_2=1|X_=x_1)P(X_1=x_1)\\ &= \frac{m}{n}*\frac{m-1}{n-1} + \frac{n-m}{n}\frac{m}{n-1}\\ &= \frac{m}{n}\\ P[X_3=1] &= \sum_{{x_1,x_2}} P(X_3=1,X_2=x_2,X_1=x_1)\\ &= \sum_{x_1,x_2} P(X_3=1|X_1=x_1,X_2=x_2)P(X_1=x_1,X_2=x_2)\\ &= \frac{m}{n} \frac{m-1}{n-1} \frac{m-2}{n-2}\\ &+ \frac{m}{n} \frac{n-m}{n-1} \frac{m-1}{n-2}\\ &+ \frac{n-m}{n} \frac{m}{n-1} \frac{m-1}{n-2}\\ &+ \frac{n-m}{n} \frac{n-m-1}{n-1} \frac{m}{n-2}\\ &= \frac{m}{n}\\ P[X_1=1,X_2=1] &= \frac{m(m-1)}{n(n-1)} \end{align*}
\begin{align*} E[X_iX_j] &= P(X_i=1,X_j=1)\\ &= \frac{m(m-1)}{n(n-1)} \\ Cov[X_i, X_j] &= E[X_i X_j]-E[X_i]E[X_j]\\ &= \frac{m(m-1)}{n(n-1)} - (\frac{m}{n})^2\\ &= \frac{m(m-n)}{n(n-1)} \end{align*}

## Problem 7(b)

\begin{align*} S &= \sum_{i=1}^n I_i\\ Var(S) &= \sum_{i=1}^k Var(I_i) + 2\sum_{i<j} Cov(I_i,I_j)\\ &= k \times (\frac{m}{n}-(\frac{m}{n})^2) + k(k-1) \frac{m(m-n)}{n(n-1)}\\ &= \frac{km}{n} (1-\frac{m}{n} + (k-1))\frac{m-n}{n(n-1)})\\ &= k\frac{m}{n}(1-\frac{m}{n})\frac{n-k}{n-1}\\ \end{align*}

# Problem 8

## Problem 8(a,b)

\begin{align*} R &= \sqrt{X^2+Y^2}\\ Q &= tan^{-1} (\frac{Y}{X})\\ \begin{vmatrix} \frac{\partial X}{\partial R} & \frac{\partial X}{\[\partial Q}\\ \frac{\partial Y}{\partial R} & \frac{\partial Y}{\[\partial Q}\\ \end{vmatrix} &= R\\ P(R,Q) &= RP(X(R), Y(Q))\\ &= \frac{R}{2\pi \sigma^2} exp^{-\frac{R^2}{2\sigma^2}}\\ &= \frac{R}{\sigma^2}exp^{-\frac{R^2}{2\sigma^2}} \times \frac{1}{2\pi}\\ &= P(R)P(Q)\\ P(R) &= \frac{R}{\sigma^2}exp^{-\frac{R^2}{2\sigma^2}}\\ \end{align*}

## Problem 8(c)

\begin{align*} P(R < k\sigma ) &= 0.5\\ \int_0^{k\sigma} \frac{\$R}{\sigma^2} exp^{-\frac{R^2}{2\sigma^2}}dR &= 0.5 exp^{-\frac{k^2}{2}}&=0.5\\ k^2 &= 2\ln(2) \end{align*}
By @Saket Choudhary in
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