Chapter 4, Integral Video Solutions, Measure, Integral and Probability

Problem 1

Find the integral of $\varphi$ over $E$ where
(a) $\varphi(x)=\operatorname{Int}(x), E=[0,10]$
(b) $\varphi(x)=\operatorname{Int}\left(x^{2}\right), E=[0,2]$
(c) $\varphi(x)=\operatorname{Int}(\sin x), E=[0,2 \pi]$
and Int denotes the integer part of a real number. (Note: many texts use the symbol $[x]$ to denote $\operatorname{Int}(x)$. We prefer to use Int for increased clarity.)

Uma Kumari

Numerade Educator

01:19

Problem 2

Suppose that $f:[0,1] \rightarrow \mathbb{R}$ is defined by letting $f(x)=0$ on the Cantor set and $f(x)=k$ for all $x$ in each interval of length $3^{-k}$ which has been removed from $[0,1] .$ Calculate $\int_{0}^{1} f \mathrm{~d} m$.

Lucas Finney

Numerade Educator

03:38

Problem 3

Prove the following Mean Value Theorem for the integral: if $a \leq f(x) \leq b$ for $x \in A$, then $a m(A) \leq \int_{A} f \mathrm{~d} m \leq b m(A)$

Alex Roush

Numerade Educator

03:11

Problem 4

Construct an example of a sequence of functions with the strict inequality as above, such that all $f_{n}$ are zero outside the interval $[0,1]$.
It is now easy to prove one of the two main convergence theorems.

Kumar Vaibhav

Numerade Educator

01:21

Problem 5

For which $\alpha$, is $f(x)=x^{\alpha \text { in }} \mathcal{L}^{1}(E)$ where (a) $E=(0,1)$; (b) $E=$ $(1, \infty) ?$

Varsha Aggarwal

Numerade Educator

03:30

Problem 7

Investigate the convergence of
$$
\int_{a}^{\infty} \frac{n^{2} x \mathrm{e}^{-n^{2} x^{2}}}{1+x^{2}} \mathrm{~d} x
$$
for $a>0$, and for $a=0$.

William Semus

Numerade Educator

04:59

Problem 8

Investigate the convergence of
$$
\int_{0}^{\infty} \frac{1}{\left(1+\frac{x}{n}\right)^{n} \sqrt[n]{x}} \mathrm{~d} x
$$
We will need the following extension of Theorem 4.12:

Priyanka Sadarangani

Numerade Educator

02:53

Problem 9

The following are variations on the above theme:
(a) For which values of $a \in \mathbb{R}$ does the power series $\sum_{n \geq 0} n^{a} x^{n}$ define an integrable function on $[-1,1] ?$
(b) Show that $\int_{0}^{\infty} \frac{x}{e^{x}-1} \mathrm{~d} x=\frac{\pi^{2}}{6}$.

Suzanne W.

Numerade Educator

00:56

Problem 10

Show that the function $f$ given by $f(x)=\frac{\sin x}{x}(x \neq 0)$ has an improper Riemann integral over $\mathbb{R}$, but is not in $\mathcal{L}^{1}$.

Amrita Bhasin

Numerade Educator

03:54

Problem 11

Show that $\int_{-\infty}^{\infty} n(x) \mathrm{d} x=1$.

Linda Hand

Numerade Educator

02:42

Problem 12

Show that $\int_{-\infty}^{\infty} c(x) \mathrm{d} x=1$
The exponential density is given by
$$
f(x)= \begin{cases}c e^{-\lambda x} & \text { if } x \geq 0 \\ 0 & \text { otherwise }\end{cases}
$$

Ahmad Reda

Numerade Educator

01:09

Problem 13

Find the constant $c$ for $f$ to be a density of probability distribution.

Tyler Gaona

Numerade Educator

01:58

Problem 14

Show that $F_{X}$ is continuous if and only if $P_{X}(\{y\})=0$ for all $y$.

Aayush Gupta

Numerade Educator

02:08

Problem 15

Find $F_{X}$ for
(a) a constant random variable $X, X(\omega)=a$ for all $\omega$
(b) $X:[0,1] \rightarrow \mathbb{R}$ given by $X(\omega)=\min \{\omega, 1-\omega\}$ (the distance to the nearest endpoint of the interval $[0,1]$ )
(c) $X:[0,1]^{2} \rightarrow \mathbb{R}$, the distance to the nearest edge of the square $[0,1]^{2}$.

Jacob Fry

Numerade Educator

00:52

Problem 16

Find the density of $Y=X^{3}$ where $f_{X}=1_{[0,1]}$.

Linh Vu

Numerade Educator

View

Problem 17

Find the expectation of
(a) a constant random variable $X, X(\omega)=a$ for all $\omega$
(b) $X:[0,1] \rightarrow \mathbb{R}$ given by $X(\omega)=\min \{\omega, 1-\omega\}$ (the distance to the nearest endpoint of the interval $[0,1]$ )
(c) $X:[0,1]^{2} \rightarrow \mathbb{R}$, the distance to the nearest edge of the square $[0,1]^{2}$

Rashmi Sinha

Numerade Educator

01:50

Problem 18

Find the mathematical expectation of a random variable with
(a) uniform distribution over the interval $[a, b]$,
(b) triangle distribution,
(c) exponential distribution.

Wendi Zhao

Numerade Educator

05:41

Problem 19

Find the characteristic function of a random variable with
(a) uniform distribution over the interval $[a, b]$,
(b) exponential distribution,
(c) Gaussian distribution.

Laurent Bergeron

Numerade Educator

01:26

Problem 20

$$
\text { Find the formula for the put option. }
$$

Gregory Higby

Numerade Educator