# 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?

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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

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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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