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?

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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

…

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