Problem set

Satisfactory Essays
ECE 302

Problem Set 9

Fall 2013

The following problems have been selected from the course text.
4.78 In a large collection of wires, the length of a wire is X, an exponential random variable with mean
5π cm. Each wire is cut to make rings of diameter 1 cm. Find the probability mass function for the number of complete rings produced by each length of wire.
4.85 The exam grades in a certain class have a Gaussian pdf with mean m and standard deviation σ. Find the constants a and b so that the random variable Y = aX + b has a Gaussian pdf with mean m and standard deviation σ .
4.86, 4.87 Let X = U n where n is a positive integer and U is a uniform random variable in the unit interval.
Find the cdf and pdf of X. Repeat for the case where U is uniform in the interval [−1, 1].
4.94 modified Let Y = α tan(πX), where X is uniformly distributed in the interval (−1/2, 1/2).
a. Show that Y is a Cauchy random variable.
b. Find the pdf of Z = 1/Y .
4.96 Find the pdf of X = − ln(1 − U ), where U is a uniform random variable in (0, 1).
4.99 modified Let X be a random variable with mean m. Compare the Chebyshev inequality and the exact probability for the event {|X − m| > c} as a function of c for the case where:
a. X is a uniform random variable in the interval [−b, b];
b. X has pdf fX (x) =

α
2

exp(−α|x|);

c. X is a zero mean Gaussian random variable with variance σ 2 .
4.100 Let X be the number of successes in n Bernoulli trials where the probability of success is p. Let
Y = X/n be the average number of successes per trial. Apply the Chebyshev inequality to the event
{|Y − p| > a}. What happens as n → ∞?
4.102

a. Find the characteristic function of the random variable X uniformly distributed over [−b, b).
b. Find the mean and variance of X by applying the moment theorem.

4.105 modified a. Show that the characteristic function of a Gaussian random variable X with mean m and variance σ 2 is
2 2
ΦX (ω) = ejmω−σ ω /2 .
b. Use the moment

You May Also Find These Documents Helpful

  • Satisfactory Essays

    Problem Set

    • 438 Words
    • 2 Pages

    Math – 213 Problem Set Solve each of the following problems. 1. A coin is tossed four times. What is the probability that at least 2 heads will occur? 2. A pair of dice is rolled. What is the probability that the sum is equal to the following? a. 5 b. 10 c. at most 9 d. at least 8 3. A chip is drawn at random from a jar containing 8 red, 2 blue, 3 green, 4 yellow, and 3 white chips. Determine the probability that it is: a. Red b. Yellow or red c. Not orange 4. In a graduating class of 300 students…

    • 438 Words
    • 2 Pages
    Satisfactory Essays
  • Satisfactory Essays

    Problem Set

    • 676 Words
    • 3 Pages

    Me1 Problem Set #2 The US College Enrollment and the “Third Law of Demand” A theorem proposed by Professors Alchian and Allen in their text, University Economics (1964) has had several rebirths of interest in the literature. The so-called “third law of demand,” or “relative price theorem,” holds that a fixed cost added to a good of varying quality causes the consumer to prefer the category considered of higher quality to the lower. Recently a number of studies, keeping this theorem in mind have…

    • 676 Words
    • 3 Pages
    Satisfactory Essays
  • Satisfactory Essays

    Problem Set

    • 1188 Words
    • 5 Pages

    Energy 1. A man climbs on to a wall that is 5 m high and gains 2200 J of potential energy. What is the mass of the man? Given: Formula: h = 5 m m = PE/gh PE = 2200 J Solution: m = 2200 J / (9.8 m/s2 x 5 m) m = 49 kg 2. Calculate the kinetic energy of a 500 kg car travelling at 50 m/s. Given: Formula: m = 500 kg KE = mv2/2 v = 50 m/s Solution: KE = 500 kg x (50 m/s)2 / 2 KE = 625 000 J Power 3. Riley climbs a flight of stairs in 3 minutes. If he…

    • 1188 Words
    • 5 Pages
    Satisfactory Essays
  • Satisfactory Essays

    Problem set

    • 1065 Words
    • 5 Pages

    1. EP Enterprises has the following income statement. How much net operating profit after taxes (NOPAT) does the firm have? Sales $1,800.00 Costs 1,400.00 Depreciation 250.00 EBIT $ 150.00 Interest expense 70.00 EBT $ 80.00 Taxes (40%) 32.00 Net income $ 48.00 a. $81.23 b. $85.50 c. $90.00 EBIT $150.00…

    • 1065 Words
    • 5 Pages
    Satisfactory Essays
  • Satisfactory Essays

    Problem Set

    • 525 Words
    • 3 Pages

    GENERAL BIOCHEMISTRY Problem Set No. 1 Answered by: Andrea Rose A. Fajardo of BSFT 2-1N 1. Calculate the pH of a dilute solution that contains a molar ratio of potassium acetate to acetic acid (pKa=4.76) of: a. 2:1 pH = 4.76 + log[2]/[1] = 4.76 + 0.3010 = 5.06 b. 1:4 pH = 4.76 + log[1]/[4] = 4.76 + (-0.6021) = 4.16 c. 6:5 pH = 4.76 + log[6]/[5] = 4.76 + 0.0792…

    • 525 Words
    • 3 Pages
    Satisfactory Essays
  • Satisfactory Essays

    Problem Set

    • 464 Words
    • 2 Pages

    BE 13.8 Vertical analysis (common-size) percentages for Vallejo Company’s sales, cost of goods sold, and expenses are listed below: VERTICAL ANALYSIS 2012 2011 2010 Sales 100.0% 100.0% 100.0% Cost of goods sold 60.5 62.9 64.8 Expenses 26.0 26.6 27.5 Net Income 13.5 10.5 7.7 Did Vallejo’s net income as a percent of sales increase, decrease, or remain unchanged over the 3 year period. Provide numerical support for your answer. Sales – Cost of Goods Sold – Expenses = Net Income 100 – 64.8…

    • 464 Words
    • 2 Pages
    Satisfactory Essays
  • Satisfactory Essays

    Practise Problem Set

    • 509 Words
    • 3 Pages

    ECSE 443- introduction to Numerical Methods in EE Practice Problem Set – Chapter 4 (Note: This problem set is for extra practice. It is not for credit, and not to be handed in) Question 1: Suppose that in a study, several measurements were made to determine a particular quantity for different values of . The collected data are summarized in the following table: 1.0000 5.8823 1.5000 2.0000 2.5000 3.0000 3.5000 4.0000 4.5000 5.0000 5.5000 6.0000 8.8823 17.8821 11.8822 14.8826 26.8825 24.8822…

    • 509 Words
    • 3 Pages
    Satisfactory Essays
  • Satisfactory Essays

    Problem set 5

    • 900 Words
    • 15 Pages

    ECONOMETRICS: PS5 PROBLEM SET 5: ESTIMATION PROBLEMS 1 We have the following variables: Y: Food expenditure in USA. X: Family income. P: Price index. Two different regressions are estimated with the following estimation results (standard errors are in brackets): Coefficient for Regression X Y/P Y / X; P 0.112 (0.003) Coefficient for P 2.462 (0.407) -0.739 (0.114) Determination coefficient 0.614 0.978 Assuming that the true equation for Y includes both X and…

    • 900 Words
    • 15 Pages
    Satisfactory Essays
  • Good Essays

    Problem Set 2

    • 922 Words
    • 4 Pages

    Problem Set 2 Name: __________________________________________ Problem Set 2 is to be completed by 11:59 p.m. (ET) on Monday of Module/Week 4. 1. The following table presents data for wages in the market for internet security professionals. (HINT: in the labor market the roles are reversed. Those who want to hire labor are the demanders. The workers enter the work force providing labor to the market place so they are the suppliers.) Wage Quantity Demanded Quantity Supplied $50,000…

    • 922 Words
    • 4 Pages
    Good Essays
  • Satisfactory Essays

    Problem Set 3

    • 294 Words
    • 2 Pages

    PROBLEM SET 3 Name: ________________________________________ Problem Set 3 is to be completed by 11:59 p.m. (ET) on Monday of Module/Week 6. 1. Data for the market for graham crackers is shown below. Calculate the elasticity of demand between the following prices. Price of crackers Quantity Demanded (per month) $3 80 $2.5 120 $2 160 $1.5 200 $1 240 $1.00 - $1.50: ___________________________________ $1.50 - $2.00: ___________________________________ $2.00 - $2.50: ___________________________________…

    • 294 Words
    • 2 Pages
    Satisfactory Essays