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Questions and Answers of Statistics

Continuation of Exercise 140 Rework parts (a) and (b). Assume that the lifetime is an exponential random variable with the same mean.
Continuation of Exercise 4-140. Rework parts (a) And (b). Assume that the lifetime is a lognormal random variable with the same mean and standard deviation.
A square inch of carpeting contains 50 carpet fibers. The probability of a damaged fiber is 0.0001. Assume the damaged fibers occur independently. (a) Approximate the probability of one or more
An airline makes 200 reservations for a flight that holds 185 passengers. The probability that a passenger arrives for the flight is 0.9 and the passengers are assumed to be independent. (a)
The steps in this exercise lead to the probability density function of an Erlang random variable X with parameters λ and r, f(x) = λr xr–1 e–λ /(r – 1) !, x > 0, r = 1, 2, .
A bearing assembly contains 10 bearings. The bearing diameters are assumed to be independent and normally distributed with a mean of 1.5 millimeters and a standard deviation of 0.025 millimeter. What
Let the random variable X denote a measurement from a manufactured product. Suppose the target value for the measurement is m. For example, X could denote a dimensional length, and the target might
The lifetime of an electronic amplifier is modeled as an exponential random variable. If 10% of the amplifiers have a mean of 20,000 hours and the remaining amplifiers have a mean of 50,000 hours,
Lack of Memory Property Show that for an exponential random variable X, P(X t1) = P(X < t2)
A process is said to be of six-sigma quality if the process mean is at least six standard deviations from the nearest specification. Assume a normally distributed measurement.(a) If a process mean is
Show that the following function satisfies the properties of a joint probability mass function.
Continuation of Exercise 5-1 Determine the following probabilities:(a) P(X < 2.5, Y < 3)(b) P(X < 2.5)(c) P(Y < 3)(d) P(X > 1.8, Y > 4.7)
Continuation of Exercise 5-1 Determine and E (X) and E(Y).
Continuation of Exercise 5-1 Determine. (a) The marginal probability distribution of the random variable X. (b) The conditional probability distribution of Y given that λ = 1.5. (c) The
Determine the value of c that makes the function f(x, y) = c (x + y) a joint probability mass function over the nine points with x = 1, 2, 3 and y = 1, 2, 3.
Continuation of Exercise 5-5 determine the following probabilities(a) P(X = 1, Y < 4)(b) P(X = 1)(c) P(Y = 2)(d) P(X < 2, Y < 2)
Continuation of Exercise 5-5 Determine E(X), E(Y), V(X), and V (Y).
Continuation of Exercise 5-5 Determine(a) The marginal probability distribution of the random variable X. (b) The conditional probability distribution of Y given that X = 1.(c) The conditional
Show that the following function satisfies the properties of a joint probability mass function.
Continuation of Exercise 5-9. Determine the following probabilities:(a) P(X < 0.5 Y < 1.5)(b) P(Y < 1.5)(c) P(X < 0.5) (d) P(X > 0.25 Y < 4.5)
Continuation of Exercise 5-9. Determine E(X) and E(Y ).
Continuation of Exercise 5-9. Determine(a) The marginal probability distribution of the random variable X.(b) The conditional probability distribution of Y given that X = 1.(c) The conditional
Four electronic printers are selected from a large lot of damaged printers. Each printer is inspected and classified as containing either a major or a minor defect. Let the random variables X and Y
In the transmission of digital information, the probability that a bit has high, moderate, and low distortion is 0.01, 0.10, and 0.95, respectively. Suppose that three bits are transmitted and that
A small-business Web site contains 100 pages and 60%, 30%, and 10% of the pages contain low, moderate, and high graphic content, respectively. A sample of four pages is selected without replacement,
A manufacturing company employs two inspecting devices to sample a fraction of their output for quality control purposes. The first inspection monitor is able to accurately detect 99.3% of the
Suppose the random variables X, Y, and Z have the following joint probability distributionDetermine the following:(a) P(X = 2)(b) P(X = 1, Y = 2)(c) P(Z (d) P(X = 1 or Z = 2)(e) E(X)
Continuation of Exercise 5-17. Determine the following:(a) P(X = 1|Y = 1)(b) P(X = 1, Y = 1|Z = 2)(c) P(X = 1|Y = 1, Z = 2)
Continuation of Exercise 5-17. Determine the conditional probability distribution of X given that Y = 1 and Z = 2.
Based on the number of voids, a ferrite slab is classified as either high medium, or low. Historically, 5% of the slabs are classified as high, 85% as medium, and 10% as low.A sample of 20 slabs is
Continuation of Exercise 5-20. Determine the following:(a) P(X = 1, Y = 17, Z = 3)(b) P(X < 1, Y = 17, Z = 3)(c) P(X < 1(d) E(X)
Continuation of Exercise 5-20. Determine the following:(a) P(X = 2, Z = 3|Y = 17)(b) P(X = 2|Y = 17)(c) E(X|Y = 17)
An order of 15 printers contains four with a graphics enhancement feature, five with extra memory, and six with both features. Four printers are selected at random, without replacement, from this
Continuation of Exercise 5-23. Determine the conditional probability distribution of X given that Y = 2.
Continuation of Exercise 5-23. Determine the following:(a) P(X = 1, Y = 2, Z =1)(b) P(X = 1, Y = 1)(c) E(X) and V(X)
Continuation of Exercise 5-23. Determine the following:(a) P(X = 1, Y = 2|Z =1)(b) P(X = 2/Y = 2)(c) The conditional probability distribution of X given that Y = 0 and Z = 3.
Four electronic ovens that were dropped during shipment are inspected and classified as containing either a major, a minor, or no defect. In the past, 60% of dropped ovens had a major defect, 30% had
Continuation of Exercise 5-27. Determine the following:(a) The joint probability mass function of the number of ovens with a major defect and the number with a minor defect.(b) The expected number of
Continuation of Exercise 5-27. Determine the following:(a) The conditional probability that two ovens have major defects given that two ovens have minor defects(b) The conditional probability that
In the transmission of digital information, the probability that a bit has high, moderate, or low distortion is 0.01, 0.04, and 0.95, respectively. Suppose that three bits are transmitted and that
Continuation of Exercise 5-30. Let X and Y denote the number of bits with high and moderate distortion out of the three transmitted, respectively. Determine the following:(a) The probability
A marketing company performed a risk analysis for a manufacturer of synthetic fibers and concluded that new competitors present no risk 13% of the time (due mostly to the diversity of fibers
Continuation of Exercise 5-32. Determine the following:(a) P(Z = 2|Y = 1, X = 10) (b) P(Z < 1|X = 10) (c) P(Z < 1, Z < = 1|X = 10) (d) E(Z|X = 10)
Determine the value of c such that the function f(x, y) = cxy for 0 < x < 3 and 0 < y < 3 satisfies the properties of a joint probability density function.
Continuation of Exercise 5-34. Determine the following:(a) P(X < 2.Y < 3)(b) P(X < 2.5)(c) P(1 < Y< 2.5) (d) P(X > 1.8, 1 < Y < 2.5)(e) E(X) (f) P(X < 0, Y < 4)
Continuation of Exercise 5-34. Determine the following:(a) Marginal probability distribution of the random variable X(b) Conditional probability distribution of Y given that X = 1.5(c) E(Y|X) =
Determine the value of c that makes the function f(x, y) = c(x = y) a joint probability density function over the range 0 < x < 3 and x < y < x + 2.
Continuation of Exercise 5-37. Determine the following:(a) P(X < 1, Y < 2)(b) P(1 < X < 2)(c) P(Y > 1)(d) P(X < 2, Y < 2)(e) E(X)
Continuation of Exercise 5-37. Determine the following:(a) Marginal probability distribution of X(b) Conditional probability distribution of Y given that X = 1(c) E(Y|X = 1)(d) P(Y > 2|X = 1)(e)
Determine the value of c that makes the function f(x, y) =cxy a joint probability density function over the range 0
Continuation of Exercise 5-40. Determine the following:(a) P(X < 1, Y< 2)(b) P(1 < X < 2)(c) P(Y > 1)(d) P(X < 2, Y< 2)(e) E(X) (f) E(Y)
Continuation of Exercise 5-40. Determine the following:(a) Marginal probability distribution of X(b) Conditional probability distribution of Y given X = 1(c) E(Y|X = 1)(d) P(Y > 2|X = 1)(e)
Determine the value of c that makes the function f(x, y) = ce-2x – 3y a joint probability density function over the range 0 < x and 0 < y < x.
Continuation of Exercise 5-43. Determine the following:(a) P(X < 1, Y < 2)(b) P(1 < X, < 2)(c) P(Y > 3)(d) P(X
Continuation of Exercise 5-43. Determine the following:(a) Marginal probability distribution of X(b) Conditional probability distribution of Y given X = 1(c) E(Y|X = 1)(d) Conditional probability
Determine the value of c that makes the function f(x, y) = ce-2x–3y a joint probability density function over the range 0 < x and x < y.
Continuation of Exercise 5-46. Determine the following:(a) P(X < 1, Y < 2)(b) P(1 < X, < 2)(c) P(Y > 2)(d) P(X < 2, Y < 2)(e) E(X)(f) E(Y)
Continuation of Exercise 5-46. Determine the following:(a) Marginal probability distribution of X(b) Conditional probability distribution of Y given X = 1(c) E(Y|X = 1)(d) P(Y
Two methods of measuring surface smoothness are used to evaluate a paper product. The measurements are recorded as deviations from the nominal surface smoothness in coded units. The joint probability
Continuation of Exercise 5-49. Determine the following:(a) P(X
Continuation of Exercise 5-49. Determine the following:(a) Marginal probability distribution of X(b) Conditional probability distribution of Y given X = 1(c) X(Y|X = 1)(d) P(Y < 0.5|X = 1)
The time between surface finish problems in a galvanizing process is exponentially distributed with a mean of 40 hours. A single plant operates three galvanizing lines that are assumed to operate
A popular clothing manufacturer receives Internet orders via two different routing systems. The time between orders for each routing system in a typical day is known to be exponentially distributed
The conditional probability distribution of Y given X = x is FY|X(y) = XE–xy for y > 0 and the marginal probability distribution of X is a continuous uniform distribution over 0 to 10.(a) Graph is
Suppose the random variables X, Y, and Z have the joint probability density function f(x, y, z) = 8xyz for 0 < x < 1, 0 < y < 1, and 0 < z < 1. Determine the following:(a) P(X < 0.5)(b) P(X < 0.5, Y
Continuation of Exercise 5-55. Determine the following:(a) P(X < 0.5|Y = 0.5)(b) P(X < 0.5, Y < 0.5|Z = 0.8)
Continuation of Exercise 5-55. Determine the following: (a) Conditional probability distribution of X given that Y = 0.5 and Z 0.8(b) P(X < 0.5|Y = 0.5, Z = 0.8)
Suppose the random variables X, Y, and Z have the joint probability density function fXYZ (x, y, z) = c over the cylinder x2 + y2 < 4 and 0 < z < 4. Determine the following.(a) The constant c so that
Continuation of Exercise 5-58. Determine the following:(a) P(X < 1|Y = 1)(b) P(X2 + Y2 < 1|Z = 1)
Continuation of Exercise 5-58. Determine the conditional probability distribution of Z given that X = 1 and Y = 1.
Determine the value of c that makes fXYZ(x, y, z) = c a joint probability density function over the region x > 0, y > 0, z > 0, and x + y + z < 1.
Continuation of Exercise 5-61. Determine the following:(a) P(X < 0.5, Y < 0.5, z < 0.5)(b) P(X < 0.5, Y < 0.5(c) P(X < 0.5)(d) E(X)
Continuation of Exercise 5-61. Determine the following:(a) Marginal distribution of X(b) Joint distribution of X and Y(c) Conditional probability distribution of X given that Y = 0.5 and Z = 0.5(d)
The yield in pounds from a day’s production is normally distributed with a mean of 1500 pounds and standard deviation of 100 pounds. Assume that the yields on different days are independent random
The weights of adobe bricks used for construction are normally distributed with a mean of 3 pounds and a standard deviation of 0.25 pound. Assume that the weights of the bricks are independent and
A manufacturer of electroluminescent lamps knows that the amount of luminescent ink deposited on one of its products is normally distributed with a mean of 1.2 grams and a standard deviation of 0.03
Determine the covariance and correlation for the following joint probability distribution:x 1 1 2 4y 3 4 5 6fXY (x, y) 1/8 1/4 1/2 1/8
Determine the covariance and correlation for the following joint probability distribution:x –1 –0.5 0.5 1y –2 –1 1 2 fXY (x, y) 1/8 ¼ ½ 1/8
Determine the value for c and the covariance and correlation for the joint probability mass function fXY (x, y) = c(x + y) for x = 1, 2, 3 and y = 1, 2, 3.
Determine the covariance and correlation for the joint probability distribution shown in Fig. 5-4(a) and described in Example 5-8.
Determine the covariance and correlation for X1 and X2 in the joint distribution of the multinomial random variables X1, X2 and X3 in with p1 = p2 = p3 = 1/3 and n = 3. What can you conclude about
Determine the value for c and the covariance and correlation for the joint probability density function fXY (x, y) = cxy over the range 0 < x < 3 and 0 < y < x.
Determine the value for c and the covariance and correlation for the joint probability density function fXY (x, y) = c over the range 0 < x < 5, 0 < y, and x – 1 < y < x + 1.
Determine the covariance and correlation for the joint probability density function fXY (x, y) = 6 X 10-6e –0.001x-0.002y over the range 0 < x and x < y from Example 5-15.
Determine the covariance and correlation for the joint probability density function fXY (x, y) = e–x–y over the range 0 < x and 0 < y.
Suppose that the correlation between X and Y is =. For constants a, b, c, and d, what is the correlation between the random variables U aX + b and V = cY + d?
The joint probability distribution is x - 1 0 0 1y 0 -1 1 0fXY (x, y) ¼ ¼ ¼ ¼ Show that the correlation between X and Y is zero, but X and Y are not independent.
Suppose X and Y are independent continuous random variables. Show that σXY = 0.
Let X and Y represent concentration and viscosity of a chemical product. Suppose X and Y have a bivariate normal distribution with σX = 4, σY = 1, μX = 2, and μY = 1. Draw a rough
Let X and Y represent two dimensions of an injection molded part. Suppose X and Y have a bivariate normal distribution with σX = 0.04, σY = 0.08, μX = 3.00, μY = 7.70, and pY = 0.
In the manufacture of electroluminescent lamps, several different layers of ink are deposited onto a plastic substrate. The thickness of these layers is critical if specifications regarding the final
Suppose that X and Y have a bivariate normal distribution with joint probability density function fXY (x, y; σX, σY, μX, μY, p). (a) Show that the conditional distribution of Y,
If X and Y have a bivariate normal distribution with p = 0, show that X and Y are independent.
Show that the probability density function fXY (x, y; σX, σY, μX, μY, p) of a bivariate normal distribution integrates to one. [Hint: Complete the square in the exponent and use
If X and Y have a bivariate normal distribution with joint probability density fXY (x, y; σX, σY, μX, μY, p), show that the marginal probability distribution of X is normal with
If X and Y have a bivariate normal distribution with joint probability density fXY (x, y; σX, σY, μX, μY, p), show that the correlation between X and Y is p. [Hint: Complete the
If X and Y are independent, normal random variables with E(X) = 0, V(X) = 4, E(Y) = 10, and V(Y) = 9.Determine the following:(a) E(2X + 3Y) (b) V(2X + 3Y)(c) P(2X + 3Y < 30)(d) P(2X + 3Y < 40)
Suppose that the random variable X represents the length of a punched part in centimeters. Let Y be the length of the part in millimeters. If E(X) = 5 and V(X) = 0.25, what are the mean and variance
A plastic casing for a magnetic disk is composed of two halves. The thickness of each half is normally distributed with a mean of 2 millimeters and a standard deviation of 0.1 millimeter and the
In the manufacture of electroluminescent lamps, several different layers of ink are deposited onto a plastic substrate. The thickness of these layers is critical if specifications regarding the final

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