# Probability Theory and Statistics

Your monthly electricity bill consists of a \$20 service charge, plus \$0.08 per kWh (kilowatt-hour)
used. Suppose the number of kWh used per month is W = 800 10X , where X ~ Binomial(40,0.5).
Calculate:
(a) E[C], the expected monthly cost of your electricity.
(b) The standard deviation of C.
(c) The probability P(C = E[C]).

Imagine that, before a measurement of some (discrete) random variable X you are asked to guess the
result. What should you choose? One strategy is to choose a number ? that is somehow ?closest? to
the most likely results of the measurement. Specifically, let the mean squared error be:
e=E[(X-?)2]
Explain the name and meaning of this quantity. Find the minimum of e as a function of ?, by
calculating the derivative de/d? and setting it equal to zero. What is the significance of your result?

Don't use plagiarized sources. Get Your Custom Essay on
Probability Theory and Statistics
Just from \$13/Page

Let X ~ Geometric(1/2). Derive a (simple!) formula for the CDF of X, and plot your result.

The CDF of the continuous random variable V is
?? 0 v<-5 FV(v)= c(v)2 -5=v<5 (a) Determine the value of the constant c required to make this CDF continuous. (b) What is P(V > 4)?
(c) What is fV (v)?
(d) CalculateE[V]andVar(V).
5. The Rayleigh random variable X has a parameter a > 0, and PDF
a2xe-a2x2/2 x=0
fX(x)= 0 otherwise
Prove that this PDF is normalized, and calculate the CDF of X.
hw05.pdf

Unformatted Attachment Preview

ENGR 3341
Probability Theory and Statistics
Homework #5, due 2/19
Prof. Gelb
Problems from textbook:
Section 3.3: problem 12, 13, 24
Section 4.1.4: problem 3
Section 4.4: problems 1, 2
Problems:
1. Your monthly electricity bill consists of a \$20 service charge, plus \$0.08 per kWh (kilowatt-hour)
used. Suppose the number of kWh used per month is W = 800 10X, where X ~ Binomial(40,0.5).
Calculate:
(a) E[C], the expected monthly cost of your electricity.
(b) The standard deviation of C.
(c) The probability P(C = E[C]).
2. Imagine that, before a measurement of some (discrete) random variable X you are asked to guess the
result. What should you choose? One strategy is to choose a number ? that is somehow ?closest? to
the most likely results of the measurement. Specifically, let the mean squared error be:
e = E[(X – ?)2 ]
Explain the name and meaning of this quantity. Find the minimum of e as a function of ?, by
calculating the derivative de/d? and setting it equal to zero. What is the significance of your result?
3. Let X ~ Geometric(1/2). Derive a (simple!) formula for the CDF of X, and plot your result.
4. The CDF of the continuous random variable V is
?
0
?
c(v 5)2
FV (v) =
?
1
v < -5 -5 = v < 5 5=v (a) Determine the value of the constant c required to make this CDF continuous. (b) What is P(V > 4)?
(c) What is fV (v)?
(d) Calculate E[V ] and Var(V ).
5. The Rayleigh random variable X has a parameter a > 0, and PDF

2 2
a2 xe-a x /2 x = 0
fX (x) =
0
otherwise
Prove that this PDF is normalized, and calculate the CDF of X.
1

attachment

Basic features
• Free title page and bibliography
• Unlimited revisions
• Plagiarism-free guarantee
• Money-back guarantee
On-demand options
• Writer’s samples
• Part-by-part delivery
• Overnight delivery
• Copies of used sources
Paper format
• 275 words per page
• 12 pt Arial/Times New Roman
• Double line spacing
• Any citation style (APA, MLA, Chicago/Turabian, Harvard)

# Our guarantees

Delivering a high-quality product at a reasonable price is not enough anymore.
That’s why we have developed 5 beneficial guarantees that will make your experience with our service enjoyable, easy, and safe.

### Money-back guarantee

You have to be 100% sure of the quality of your product to give a money-back guarantee. This describes us perfectly. Make sure that this guarantee is totally transparent.

### Zero-plagiarism guarantee

Each paper is composed from scratch, according to your instructions. It is then checked by our plagiarism-detection software. There is no gap where plagiarism could squeeze in.

### Free-revision policy

Thanks to our free revisions, there is no way for you to be unsatisfied. We will work on your paper until you are completely happy with the result.

Your email is safe, as we store it according to international data protection rules. Your bank details are secure, as we use only reliable payment systems.

### Fair-cooperation guarantee

By sending us your money, you buy the service we provide. Check out our terms and conditions if you prefer business talks to be laid out in official language.

## Calculate the price of your order

550 words
We'll send you the first draft for approval by September 11, 2018 at 10:52 AM
Total price:
\$26
The price is based on these factors:
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}