Paper
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 ...
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MATH505a 2014 Spring
Paper
http://www-bcf.usc.edu/~mathgp/quals/20141/20141_505a.pdf
Problem 1
Define
$$I_i = \begin{cases} 1 & \text{Game $i$ and Game $i+1$ are won},\\ 0 & \text{otherwise} \end{cases}$$Thus,
$$ \begin{align*} R &= \sum_{i=1}^{m-1}I_i\\ ER &= \sum_{i=1}^{m-1}EI_i\\ Var(R) &= \sum_{i ...read moreMATH505a 2013 Spring
Paper
http://www-bcf.usc.edu/~mathgp/quals/20061/505aspring06.pdf
Problem 1
Given: \(E[X|Y]=X\ and \ E[Y|X]=X\)
To Show:
Part (a): \(P(X=Y)=1\)
$$ E[Y]=E[E[Y|X]]=E[X]=\mu_x $$Thus, \(\mu_y=\mu_x\)
Also,
$$ \begin{align} Cov(X,Y) &=E[XY ...read moreMATH-541a 2014 Spring
Source
http://www-bcf.usc.edu/~mathgp/quals/20141/20141_541a.pdf
Problem 1
Given:
\(x_{i,1}, x_{i,2}\) : Known values For \(1 \leq i \leq n\): \(Z_i = \beta_1x_{i,1} +\epsilon_i $ and $Y_i = \beta_1x_{i,1} +\beta_2x_{i,2} + \epsilon_i\) \(\epsilon_i \sim N(0,1)\)
(a)
Given \({\bf{Z ...
read moreMATH505a 2006 Spring
Paper
http://www-bcf.usc.edu/~mathgp/quals/20061/505aspring06.pdf
Problem 1
Define an indicator variable \(I_i\) as:
$$ I_i = \begin{cases} 1 & \text{if $i^{th}$ and $(i+1)^{th}$ cards are different (H,T) or (T,H)},\\ 0 & \text{otherwise} \end{cases} $$Now the number of runs in a ...
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