Quiz 1 Solution

1.  Let $O=\{1,3,5,\dots \}$ denote the set of odd natural numbers.  Find $\sum _{k\in O}{3}^{-k}$.

Solution:   An odd natural number is of the form $2m+1$ for some natural number $m$.  Thus

$\sum _{k\in O}{3}^{-k}=\sum _{m=0}^{\mathrm{\infty }}{3}^{-(2m+1)}$

$=3^{-1}\sum _{m=0}^{\mathrm{\infty }}{3}^{-2m}$

$=\frac{1}{3}\sum _{m=0}^{\mathrm{\infty }}{\left(3^{-2}\right)}^m$  (observe the geometric series)

$=\frac{1}{3}\frac{1}{1-\frac{1}{9}}$

$=\frac{3}{8}=0.375$

2.  Let $f(x)=\frac{e^{2x}}{e^{2x}+e^{-2x}}$.  Determine $f^{\prime }(0)$, i.e., the value of the derivative of $f(x)$ at $x=0$.

Solution:  Applying the differentiation rule for the derivative of a quotient, and the chain rule when needed, we get

$f^{\prime }(x)=\frac{2e^{2x}(e^{2x}+e^{-2x})-e^{2x}(2e^{2x}-2e^{-2x})}{{(e^{2x}+e^{-2x})}^2}$.

Plugging in $x=0$ yields

$f^{\prime }(0)=\frac{2e^0(e^0+e^0)-e^0(2e^0-2e^0)}{{(e^0+e^0)}^2}$ $=\frac{2\cdot (1+1)-1\cdot (2-2)}{(1+1{)}^2}=1$.

3.  Let $\mathcal{R}=\{(x,y)\in {\mathbb{R}}^2:0\le y\le x\}$.  Find ${\iint }_{\mathcal{R}}{e}^{-(x+y)}\text{d}y\text{d}x$.

Solution:  the region of integration is an infinite wedge in the first quadrant.  We find

${\iint }_{\mathcal{R}}{e}^{-(x+y)}\text{d}y\text{d}x={\int }_{x=0}^{\mathrm{\infty }}\left({\int }_{y=0}^x{e}^{-x}{e}^{-y}\text{d}y\right)\text{d}x$

$={\int }_{x=0}^{\mathrm{\infty }}{e}^{-x}\left({\int }_{y=0}^x{e}^{-y}\text{d}y\right)\text{d}x$

$={\int }_{x=0}^{\mathrm{\infty }}{e}^{-x}\left({(-e^{-y})|}_0^x\right)\text{d}x$

$={\int }_{x=0}^{\mathrm{\infty }}{e}^{-x}(1-e^{-x})\text{d}x$

$={\int }_{x=0}^{\mathrm{\infty }}{e}^{-x}\text{d}x-{\int }_{x=0}^{\mathrm{\infty }}{e}^{-2x}\text{d}x$

$=1-\frac{1}{2}=\frac{1}{2}=0.5$

The remaining questions pertain to a situation in which an urn initially contains three balls:   one of them is labelled "0" and two of them are labelled "1," as illustrated.

4.  A random experiment consists of selected two balls in succession (without replacement) from the urn, noting the number of each ball drawn in order of occurrence.  Specify a minimal sample space for this experiment.  (Minimal means that outcomes that can never occur are not included in the sample space.)  Hint: to specify a set, enclose the elements of the set in curly brackets (braces) { ... }.

Solution:  $\{(0,1),(1,0),(1,1)\}$  (each outcome is an ordered pair)

5.  Suppose the experiment is modified so that the ball is replaced in the urn after the first selection.  What is a minimal sample space now?

Solution:  $\{(0,0),(0,1),(1,0),(1,1)\}$

6.  Suppose the experiment is modified so that balls are sampled (without replacement) until the ball labelled "0" is drawn, and the number of balls that had to be drawn is recorded.  What is the minimal sample space now?

Solution:  $\{1,2,3\}$  (outcomes are numbers)

7.  Repeat the previous question under the assumption that the ball is replaced in the urn after each selection.

Solution:  $\{1,2,3,\dots \}$.