EXERCISES Section 4.1 (1-10)

1

The current in a certain circuit as measured by an ammeter is a continuous random variable $X$ with the following density function:

$$f\left(x\right)=\{\begin{array}{cc}.075x+.2& 3\le x\le 5\\ 0& \text{ otherwise }\end{array}$$

a. Graph the pdf and verify that the total area under the density curve is indeed 1 .

b. Calculate $P\left(X\le 4\right)$ . How does this probability compare to $P\left(X<4\right)$ ?

c. Calculate $P\left(3.5\le X\le 4.5\right)$ and also $P\left(4.5<X\right)$ .

2

Suppose the reaction temperature $X$ (in ${}^{\circ }\mathrm{C}$ ) in a certain chemical process has a uniform distribution with $A=-5$ and $B=5$ .

a. Compute $P\left(X<0\right)$ .

b. Compute $P\left(-2.5<X<2.5\right)$ .

c. Compute $P\left(-2\le X\le 3\right)$ .

d. For $k$ satisfying $-5<k<k+4<5$ , compute $P\left(k<X<k+4\right)$ .

3

The error involved in making a certain measurement is a continuous rv $X$ with pdf

$$f\left(x\right)=\{\begin{array}{cc}.09375\left(4-x^2\right)& -2\le x\le 2\\ 0& \text{ otherwise }\end{array}$$

a. Sketch the graph of $f\left(x\right)$ .

b. Compute $P\left(X>0\right)$ .

c. Compute $P\left(-1<X<1\right)$ .

d. Compute $P\left(X<-.5\text{or}X>.5\right)$ .

4

Let $X$ denote the vibratory stress (psi) on a wind turbine blade at a particular wind speed in a wind tunnel. The article “Blade Fatigue Life Assessment with Application to VAWTS” (J. of Solar Energy Engr., 1982: 107-111) proposes the Rayleigh distribution, with pdf

$$f\left(x;\theta \right)=\{\begin{array}{cc}\frac{x}{{\theta }^2}\cdot e^{-x^2/\left(2{\theta }^2\right)}& x>0\\ 0& \text{ otherwise }\end{array}$$

as a model for the $X$ distribution.

a. Verify that $f\left(x;\theta \right)$ is a legitimate pdf.

b. Suppose $\theta =100$ (a value suggested by a graph in the article). What is the probability that $X$ is at most 200? Less than 200 ? At least 200 ?

c. What is the probability that $X$ is between 100 and 200 (again assuming $\theta =100$ )?

d. Give an expression for $P\left(X\le x\right)$ .

5

A college professor never finishes his lecture before the end of the hour and always finishes his lectures within 2 min after the hour. Let $X=$ the time that elapses between the end of the hour and the end of the lecture and suppose the pdf of $X$ is

$$f\left(x\right)=\{\begin{array}{cc}k{x}^2& 0\le x\le 2\\ 0& \text{ otherwise }\end{array}$$

a. Find the value of $k$ and draw the corresponding density curve.

• Hint: Total area under the graph of $f\left(x\right)$ is 1.

b. What is the probability that the lecture ends within 1 min of the end of the hour?

c. What is the probability that the lecture continues beyond the hour for between 60 and 90 sec?

d. What is the probability that the lecture continues for at least 90 sec beyond the end of the hour?

6

The actual tracking weight of a stereo cartridge that is set to track at $3\mathrm{g}$ on a particular changer can be regarded as a continuous rv $X$ with pdf

$$f\left(x\right)=\{\begin{array}{cc}k\left[1-{\left(x-3\right)}^2\right]& 2\le x\le 4\\ 0& \text{ otherwise }\end{array}$$

a. Sketch the graph of $f\left(x\right)$ .

b. Find the value of $k$ .

c. What is the probability that the actual tracking weight is greater than the prescribed weight?

d. What is the probability that the actual weight is within $.25\mathrm{g}$ of the prescribed weight?

e. What is the probability that the actual weight differs from the prescribed weight by more than $.5\mathrm{g}$ ?

7

The article “Second Moment Reliability Evaluation vs. Monte Carlo Simulations for Weld Fatigue Strength” (Quality and Reliability Engr. Intl., 2012: 887-896) considered the use of a uniform distribution with $A=.20$ and $B=4.25$ for the diameter $X$ of a certain type of weld $\left(\mathrm{mm}\right)$ .

a. Determine the pdf of $X$ and graph it.

b. What is the probability that diameter exceeds $3\mathrm{mm}$ ?

c. What is the probability that diameter is within $1\mathrm{mm}$ of the mean diameter?

d. For any value $a$ satisfying $.20<a<a+1<4.25$ , what is $P\left(a<X<a+1\right)$ ?

8

In commuting to work, a professor must first get on a bus near her house and then transfer to a second bus. If the waiting time (in minutes) at each stop has a uniform distribution with $A=0$ and $B=5$ , then it can be shown that the total waiting time $Y$ has the pdf

$$f\left(y\right)=\{\begin{array}{ll}\frac{1}{25}y& 0\le y<5\\ \frac{2}{5}-\frac{1}{25}y& 5\le y\le 10\\ 0& y<0\text{ or }y>10\end{array}$$

a. Sketch a graph of the pdf of $Y$ .

b. Verify that ${\int }_{-\infty }^{\infty }f\left(y\right)dy=1$ .

c. What is the probability that total waiting time is at most 3 min?

d. What is the probability that total waiting time is at most $8\min$ ?

e. What is the probability that total waiting time is between 3 and 8 min?

f. What is the probability that total waiting time is either less than 2 min or more than 6 min?

9

Based on an analysis of sample data, the article “Pedestrians” Crossing Behaviors and Safety at Unmarked Roadways in China” (Accident Analysis and Prevention, 2011: 1927-1936) proposed the pdf $f\left(x\right)=.15e^{-.15\left(x-1\right)}$ when $x\ge 1$ as a model for the distribution of $X=$ time (sec) spent at the median line.

a. What is the probability that waiting time is at most 5 sec? More than 5 sec?

b. What is the probability that waiting time is between 2 and 5 sec?

10

A family of pdf’s that has been used to approximate the distribution of income, city population size, and size of firms is the Pareto family. The family has two parameters, $k$ and $\theta$ , both $>0$ , and the pdf is

$$f\left(x;k,\theta \right)=\{\begin{array}{cc}\frac{k\cdot {\theta }^k}{x^{k+1}}& x\ge \theta \\ 0& x<\theta \end{array}$$

a. Sketch the graph of $f\left(x;k,\theta \right)$ .

b. Verify that the total area under the graph equals 1 .

c. If the rv $X$ has pdf $f\left(x;k,\theta \right)$ , for any fixed $b>\theta$ , obtain an expression for $P\left(X\le b\right)$ .

d. For $\theta <a<b$ , obtain an expression for the probability $P\left(a\le X\le b\right)$ .