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Using a Mass Function
Let \(X\) be a random variable with probability mass function
\(p(n) = \frac{1}{c^n}\quad \text{for } n=2,3,4,\cdots\)
and \(p(x)=0\) otherwise.
- Find \(c.\)
- Compute the probability that \(X\) is even.
Magic Die (or is it rigged?)
A magician claims to have a magic die. If the die is rolled and lands on an even number, then the next time the die is rolled it will land on an odd number (and vice versa). So, as the die is rolled it will alternate perfectly between even and odd numbers (or so the magician claims).
You, being skeptical, figure there’s a 1 percent chance that the die is magical and a 99 percent chance that it’s just an ordinary fair die. You then ask the magician to “prove” the die is magical by rolling it some number of times.
How many successfully alternating rolls will it take for you to think there’s a 99 percent chance the die is magical (or, more likely, that it’s rigged in some way so it always alternates)?
Flipping Coins and Independence
An experimenter has two fair coins and one biased coin. The biased coin lands on heads with probability 3/4.
The experimenter randomly selects one of the three coins and flips it until they get heads.
Let \(A\) be the event that the experimenter flipped the biased coin.
Let \(B\) be the event that it took the experimenter an even number of flips to get heads.
Are events \(A\) and \(B\) independent?
Coin flipping game
Your friend challenges you to a game in which you flip a fair coin until you get heads. If you flip an even number of times, you win. Let \(A\) be the event that you win. Let \(B\) be the event that you flip the coin 3 or more times. Let \(C\) be the event that you flip the coin 4 or more times.
- Compute \(\mathbb{P}(A)\).
- Are \(A\) and \(B\) independent?
- Are \(A\) and \(C\) independent?
Diffusion and Brownian motion
Let \(B_t\) be a standard Brownian Motion starting from zero and define
\[ p(t,x) = \frac1{\sqrt{2\pi t}}e^{-\frac{x^2}{2t} } \]
Given any \(x \in \mathbf R \), define \(X_t=x + B_t\) . Of course \(X_t\) is just a Brownian Motion stating from \(x\) at time 0. Fixing a smooth, bounded, compactly supported function \(f:\mathbf R \rightarrow \mathbf R\), we define the function \(u(x,t)\) by
\[u(x,t) = \mathbf E_x f(X_t)\]
where we have decorated the expectation with the subscript \(x\) to remind us that we are starting from the point \(x\).
- Explain why \[ u(x,t) = \int_{\infty}^\infty f(y)p(t,x-y)dy\]
- Show by direct calculation using the formula from the previous question that for \(t>0\), \(u(x,t)\) satisfies the diffusion equation
\[ \frac{\partial u}{\partial t}= c\frac{\partial^2 u}{\partial x^2}\]
for some constant \(c\). (Find the correct \(c\) !) - Again using the formula from part 1), show that
\[ \lim_{t \rightarrow 0} u(t,x) = f(x)\]
and hence the initial condition for the diffusion equation is \(f\).
A PDE example
Observe that for \(k=0,1,\dots\)
\[\phi_k(x) = \sin(\pi k x/2) \]
form an orthonormal basis of function for \(L^2([0,2]\) with \(\phi(0)=\phi(2)=0\). Here the inner-product of two functions in \(f,g \in L^2([0,2]\) is
\[\langle f,g\rangle =\int_0^2 f(x)g(x) dx\]
Define the operator \(L\) acting on a function \(\phi(x)\) by
\[L\phi(x)=\frac12 \frac{\partial^2 \phi}{\partial^2x}(x) – 5 \phi(x)\]
To solve the equation
\[ \frac{\partial u}{\partial t}(x,t) = (L u)(x,t) \]
with
\[ u(0,t)=u(2,t)=0 \qquad\text{and}\qquad u(x,0)=F(x) \]
assume that \(u(x,t)\) takes the from
\[u(x,t)=\sum_{k=0}^\infty a_k(t) \phi_k(x)\]
Find the equations for the \(a_k\) and solve then find an expression for \(u(x,t)\).
Taylor Series
Write in first 4 terms of the Taylor series expansion around \(x=0\) for the following functions:
- \(\log(1+x)\)
- \(e^{a x}\)
- \[\frac{1}{1-x}\]
Areas and circles
Let \(C_1\) be the circle of radius 2 centered at the origin \((0,0)\). Let \(C_2\) be the circle of radius 1 centered at the origin \((1,0)\).
- Draw a picture of matching the above description and shade the region inside of \(C_1\) but outside of \(C_2\).
- What is the area of the region you shaded ?
- What is the ratio of the shaded area to the area inside \(C_1\) ?
- What is the ratio of the area inside \(C_2\) to the area inside \(C_1\) ?
Arithmetic sum
Show that
\[\sum_{j=1}^n j =\frac{1}{2}n(n+1)\]
Hint: notice that \(1+n=n+1\), \(2+(n-1)=n+1\), \(3+(n-2)=n+1\) and so on. How many such pairings exist ?
Calculus: Differentiation
Perform the following differentiation:
- \[\frac{d\ }{dx} \Big(x^4\Big) \]
- \[\frac{d\ }{dx} \Big(x^2 \exp(-x)\Big) \]
- \[\frac{d\ }{dx} \Big(\ln(x^2) \Big) \]
Calculus: Infinite Sums
Evaluate the following infinite sums:
- \[ \sum_{k=1}^\infty \big(\frac14\big)^k\]
- \[ \sum_{k=1}^\infty \frac{3^k}{k!}\]
Calculus: Areas
Draw a picture of the region of the \(xy\)-plane were both x and y are between 0 and 1 and \(y \geq x^2\). Find the area of this region
Calculus : Exponentials integrals
Do the following integrals:
- \[ \int_0^1 x^3 dx\]
- (*) Hint: integrate by parts. \[\int_0^\infty x \exp(-x) dx\]
- (**) \[\int_0^\infty \exp(-\frac12 x^2) dx\] Try looking at \[\Big(\int_0^\infty \exp(-\frac12 x^2) dx\Big)\Big(\int_0^\infty \exp(-\frac12 y^2) dy\Big)=\int_0^\infty \int_0^\infty \exp(-\frac12 x^2) \exp(-\frac12 y^2)dx dy\] and then changing to polar coordinates. How des this help you figure out the original question ?