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Problem Set #8 for ECE 313: Fall 2003 at University of Illinois, Assignments of Statistics

University of Illinois - Urbana-ChampaignStatistics

Problem set #8 for the electrical and computer engineering (ece) 313 course at the university of illinois, due on october 31, 2003. The problems involve calculating cumulative probability distribution functions (cdfs), finding probability densities, and determining probabilities of random variables.

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Uploaded on 03/10/2009

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University of Illinois Fall 2003
ECE 313: Problem Set #8
Assigned: Friday, October 24, 2003
Due: Friday, October 31, 2003
1. Which of the following are valid cumulative probability distribution functions? For those
that are not valid cdfs, state at least one property of the cdf which is not satisfied. For
those that are valid cdfs, compute P{|X|>0.5}.
(a) FX(u) =
0u < 1
2u−u21≤u≤2
1u > 2
(b) FX(u) =
0.5e2uu < 0
1−0.25e−3uu≥0
(c) FX(u) = (0.5e2uu≤0
1−0.25e−3uu > 0
2. The amount of bread (in hundreds of pounds) that a bakery sells in a day is a random
variable Xwith probability density function
fX(u) =
cu 0≤u < 3
c(6 −u) 3 ≤u < 6
0 otherwise
(a) Find the value of the constant c.
(b) Compute the cdf FX(u) of X.
(c) Show that the function FX(u) computed in part (b) satisfies all the four properties of
cdfs given in [Ross, pp. 166-167].
(d) What is the probability that the number of pounds of bread sold in a single day will
be (i) more than 300 pounds? (ii) between 150 and 900 pounds?
(e) If Aand Bare the events in (i) and (ii), respectively, are these events independent?
3. A number is randomly chosen (that is, with uniform distribution) from the interval (0,1).
What is the probability that
(a) its first decimal digit will be a 1?
(b) its second decimal digit will be a 2?
(c) the first decimal digit of its square root will be a 3?
4. Suppose that the duration in minutes of long-distance telephone conversations follows an
exponential probability density function
fX(u) = (1
5e−u/5u > 0
0 otherwise
Find the probability that the duration of the conversation
pf2

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University of Illinois Fall 2003

ECE 313: Problem Set

Assigned: Friday, October 24, 2003

Due: Friday, October 31, 2003

  1. Which of the following are valid cumulative probability distribution functions? For those that are not valid cdfs, state at least one property of the cdf which is not satisfied. For those that are valid cdfs, compute P {|X| > 0. 5 }.

(a) FX (u) =

 



0 u < 1 2 u − u^2 1 ≤ u ≤ 2 1 u > 2

(b) FX (u) =

  

  1. 5 e^2 u^ u < 0 1 − 0. 25 e−^3 u^ u ≥ 0

(c) FX (u) =

{

  1. 5 e^2 u^ u ≤ 0 1 − 0. 25 e−^3 u^ u > 0

  2. The amount of bread (in hundreds of pounds) that a bakery sells in a day is a random variable X with probability density function

fX (u) =

 



cu 0 ≤ u < 3 c(6 − u) 3 ≤ u < 6 0 otherwise

(a) Find the value of the constant c. (b) Compute the cdf FX (u) of X. (c) Show that the function FX (u) computed in part (b) satisfies all the four properties of cdfs given in [Ross, pp. 166-167]. (d) What is the probability that the number of pounds of bread sold in a single day will be (i) more than 300 pounds? (ii) between 150 and 900 pounds? (e) If A and B are the events in (i) and (ii), respectively, are these events independent?

  1. A number is randomly chosen (that is, with uniform distribution) from the interval (0, 1). What is the probability that

(a) its first decimal digit will be a 1? (b) its second decimal digit will be a 2? (c) the first decimal digit of its square root will be a 3?

  1. Suppose that the duration in minutes of long-distance telephone conversations follows an exponential probability density function

fX (u) =

{ (^1) 5 e

−u/ (^5) u > 0 0 otherwise

Find the probability that the duration of the conversation

(a) will exceed 5 minutes? (b) will be less than 6 minutes? (c) will be between 5 and 6 minutes? (d) will be less than 6 minutes, given that it was greater than 5 minutes?

  1. Prove that the CDF FX (u) of a random variable X is right continuous, that is, FX (u+) = FX (u).
  2. Let X be a discrete RV taking on values { 1 , 2 , · · · , n} with probabilities {p 1 , p 2 , · · · , pn}. From the lectures, you know that the pdf of X can be written as

fX (u) =

∑^ n

j=

pj δ(u − j)

and the CDF as FX (u) =

∫ (^) u

−∞

fx(t)dt

(a) Sketch fX (u) and FX (u). (b) Prove that E[X] =

∑n j=1 jpj^ =^

∫ (^) ∞ 0 [1^ −^ Fx(u)]du

  1. Let X 1 , X 2 ,... , Xn be independent RVs with the same CDF FX (u) and pdf fX (u). Let Y = max{X 1 , X 2 ,... , Xn}. Determine FY (u) and fY (u) in terms of FX (u) and fX (u).