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Hitting Zero
Consider a random walk making \(2n\) steps, and let \(T\) be the first
return to its starting point, that is
\[T =\ min\{1 ≤ k ≤ 2n : S_k = 0\},\]
and \(T = 0\) if the walk does not return to zero in the first \(2n steps.
Show that for all \( 1 ≤ k ≤ n\) we have,
\[P(T = 2k) =\frac1{2k − 1} \begin{pmatrix} 2k\\k\end{pmatrix} 2^{-2k}\]
[Meetster Ex 3.3.2]
Geometric probability
in each case, consider a point picked uniformly randomly from the interior of the region. Find the probability density function for the \(x\)-coordinate.
- The square with corner : \( (-2,0), (0,2), (2,0), (0,-2) \)
- The triangle with corners: \( (-2,0), (1,0), (0,2) \)
- The polygon with corners: \( (0,2),(2,1), (1,-1), (-1,0)\)
[Pitman p277, # 12]
Expected value of Random Walk
Consider a random walk \(S_i\), which begins at zero, and makes \(2n\) steps.
Let \(k < n\). Show that
- \(\mathbf{E}\Big( |S_{2k}| \ \Big|\ |S_{2k−1}| = r\Big) = r\);
- \( \mathbf{E}\Big(|S_{2k+1}| \ \Big|\ |S_{2k}| = r\Big) = 1 \text{ if } r = 0, \text{ and } \mathbf{E}\Big(|S_{2k+1}| \ \Big|\ |S_{2k}| = r\Big) = r\text{ otherwise}.\)
Approximating sums of uniform random variables
Suppose \(X_1,X_2,X_3,X_4\) are independent uniform \((0,1)\) and we set \(S_4=X_1+X_2+X_3+X_4\). Use the normal approximation to estimate \(\mathbf{P}( S_4 \geq 3) \).
probability density example
Suppose \(X\) takes values in\( (0,1) \) and has a density
\[f(x)=\begin{cases}c x^2 (1-x)^2 \qquad &x\in(0,1)\\ 0 & x \not \in (0,1)\end{cases}\]
for some \(c>0\).
- Find \( c \).
- Find \(\mathbf{E}(X)\).
- Find \(\mathrm{Var}(X) \).
Infinite Mean
Suppose that \(X\) is a random variable whose density is
\[f(x)=\frac{1}{2(1+|x|)^2} \quad x \in (-\infty,\infty)\]
- Draw a graph of \(f(x)\).
- Find \(\mathbf{P}(-1 <X<2)\).
- Find \(\mathbf{P}(X>1)\).
- Is \(\mathbf{E}(X) \) defined ? Explain.
Stuffing Envelopes
You write a stack of thank you cards for people who gave you presents for your birthday. You address all of the envelopes but before you can stuff them you are called away. A friend tying to help you see the stack of cards and stuffs them in the envelops. Unfortunately they did not realize that each card was personalized and just stick them in the envelops randomly. Assuming there were \(n\) cards and \(n\) envelops, let \(X_{n}\) be the number of cards in the correct envelope.
- Find \(\mathbf{E} (X_n)\).
- Show that the variance of \(X_n\) is the same as \(\mathbf{E} (X_n)\).
- Is there any common distribution which has the above statistics ?
- (**) Show that
\[ \lim_{n\rightarrow \infty} \mathbf{P}(X_n =m) = \frac{1}{e\, m!}\]
