Questions and Answers of Applied Statistics and Probability

The following data are direct solar intensity measurements (watts/m2) on different days at a location in southern Spain: 562, 869, 708, 775, 775, 704, 809, 856, 655, 806, 878, 909, 918, 558, 768,
Suppose that you add 10 to all of the observations in a sample. How does this change the sample mean? How does it change the sample standard deviation?
Can the sample standard deviation be equal to zero? If so, give an example.
For any set of data values, is it possible for the sample standard deviation to be larger than the sample mean? If so, give an example.
Will the sample mean always be the most frequently occurring data value in the sample?
Will exactly half of the observations in a sample fall below the mean?
Will the sample mean always correspond to one of the observations in the sample?
Use the properties of moment generating functions to show that a sum of p independent normal random variables with means μi and variances σi2 for i = 1,2, ...., p has a normal distribution.
This exercise extends the hypergeometric distribution to multiple variables. Consider a population with N items of k different types. Assume that there are N1items of type 1, N2items of type
Suppose that the range of the continuous variables X and Y is 0 < x < a and 0 < y < b. Also suppose that the joint probability density function fXY(x, y) = g(x)h(y), where g(x) is a
Suppose that the joint probability function of the continuous random variables X and Y is constant on the rectangle 0 < x < a, 0 < y < b. Show that X and Y are independent.
Show that if X1, X2,…, Xp are independent random variables and Y = c1X1 + c2X2 + ... + cpXp, V (Y ) = c12V (X1) + c22V (X2) +…+ c2pV1 (Xp) You may assume that the random variables are continuous.
Show that if X1, X2,…, Xp are independent, continuous random variables, P(X1 ∈ A1, X2 ∈ A2,…, Xp ∈ Ap) = P(X1 ∈ A1)P(X2∈ A2) … P(Xp ∈ Ap) for any regions A1, A2,…, Ap in the
Use moment generating functions to determine the normalized power [E(X4)]1/4 from a cycling power meter when X has a normal distribution with mean 200 and standard deviation 20 Watts.
The intensity (mW/mm2) of a laser beam on a surface theoretically follows a bivariate normal distribution with maximum intensity at the center, equal variance σ in the x and y directions, and zero
The power in a DC circuit is P = I2 /R where I and R denote the current and resistance, respectively. Suppose that I is normally distributed with mean of 200 mA and standard deviation 0.2 mA and R is
Determine the value of c such that the function f(x,y) = cx2y for 0 < x < 3 and 0 < y < 2 satisfies the properties of a joint probability density function.Determine the following:(a) P(X
Suppose X has a lognormal distribution with parameters θ and ω. Determine the probability density function and the parameters values for Y = Xγ for a constant γ > 0. What is the name of this
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
An order of 15 printers contains 4 with a graphics- enhancement feature, 5 with extra memory, and 6 with both features. Four printers are selected at random, without replacement, from this set. Let
Ifdetermine E(X), E(Y), V(X), V(Y), and Ï by reorganizing the parameters in the joint probability density function. w (x, y) = 10.72 (x-1) - 1.6(x– 1)(y – 2) + (y – 2)}° | exp 1.2π
Suppose that X and Y have a bivariate normal distribution with σX = 4, σY = 1, /, μY = 4, and ρ = − 0.2. Draw a rough contour plot of the joint probability density function.
The joint distribution of the continuous random variables X, Y, and Z is constant over the region x2 + y2 ≤ 1, 0< z < 4. Determine the following:(a) P(X2 + Y2 ≤ 0.5) (b) P(X2 + Y2 ≤
To evaluate the technical support from a computer manufacturer, the number of rings before a call is answered by a service representative is tracked. Historically, 70% of the calls are answered in
The backoff torque required to remove bolts in a steel plate is rated as high, moderate, or low. Historically, the probability of a high, moderate, or low rating is 0.6, 0.3, or 0.1, respectively.
The percentage of people given an antirheumatoid medication who suffer severe, moderate, or minor side effects are 10, 20, and 70%, respectively. Assume that people react independently and that 20
Show that the following function satisfies the properties of a joint probability mass function:Determine the following:(a) P(X <0.5, Y <1.5) (b) P(X ‰¤ 1)(c) P(X
Suppose that Xi has a normal distribution with mean μi and variance σi 2 , i = 1, 2. Let X1 and X2 be independent.(a) Find the moment-generating function of Y = X1 + X1.(b) What is the distribution
Let X1, X2 X3,..., be independent exponential random variables with parameter λ.(a) Find the moment-generating function of Y = X1 + X2 + …+ Xr .(b) What is the distribution of the random variable
A random variable X has the gamma distribution0 F(x) = говr Aryle X>0 " style="" class="fr-fic fr-dib">(a) Show that the moment-generating function of X is0 F(x) = говr Aryle X>0 Ma()-(1-£)"
A random variable X has the exponential distributionShow that the moment-generating function of X is(b) Find the mean and variance of X. f(x) = le, x>0 10-(1-) Mx(t) =
The continuous uniform random variable X has density function(a) Show that the moment-generating function is(b) Use MX t ( ) to find the mean and variance of X. F(x) =, asiSB asx
A continuous random variable X has the following probability distribution: f (x) = 4xe−2x , x > 0(a) Find the moment-generating function for X.(b) Find the mean and variance of X.
The chi-squared random variable with k degrees of freedom has moment-generating function MX (t)=(1−2t)− k / 2. Suppose that X1 and X2 are independent chi-squared random variables with k1 and k2
A random variable X has the Poisson distribution(a) Show that the moment-generating function is(b) Use MX (t) to fi nd the mean and variance of the Poisson random variable. f(x) = х%3D0,1,... x!
A random variable X has the discrete uniform distribution(a) Show that the moment-generating function is(b) Use MX (t) to fi nd the mean and variance of X. =, x=1,2,.m f(x) т e (1-ет) Mx(t) =
Power meters enable cyclists to obtain power measurements nearly continuously. The meters also calculate the average power generated over a time interval. Professional riders can generate 6.6 watts
The computational time of a statistical analysis applied to a data set can sometimes increase with the square of N, the number of rows of data. Suppose that for a particular algorithm, the
Derive the probability density function for a lognormal random variable Y from the relationship that Y = exp(W) for a normal random variable W with mean θ and variance ω2.
An aircraft is flying at a constant altitude with velocity magnitude r1 (relative to the air) and angle θ1 (in a two-dimensional coordinate system). The magnitude and direction of the wind are r2
The random variable X has the probability distributionDetermine the probability distribution of Y = (X €“ 2)2. 0srS4 х S«(x)=.
Suppose that X has the probability distribution fX (x) = 1, 1 ≤ x ≤ 2 Determine the probability distribution of the random variable Y = eX.
The velocity of a particle in a gas is a random variable V with probability distribution fV (v) = av2 e–bv v > 0 where b is a constant that depends on the temperature of the gas and the mass of
A random variable X has the probability distribution fX (x) = e–x, x ≥ 0 Determine the probability distribution for the following:(a) Y = X2(b) Y = X ½(c) Y = ln X
Suppose that X has a uniform probability distribution fX (x) ,= 1, 0 ≤ x ≤ 1 Show that the probability distribution of the random variable Y = –2 X is chi-squared with two degrees of freedom.
Suppose that X is a continuous random variable with probability distribution (a) Determine the probability distribution of the random variable Y = 2X + 10.(b) Determine the expected value of Y.
Let X be a binomial random variable with p = 0.25 and n = 3. Determine the probability distribution of the random variable Y = X2.
Suppose that X is a random variable with probability distribution fX (x) = 1/4, x = 1 2 3 4 Determine the probability distribution of Y = 2X + 1.
Consider the perimeter of a part in Example 5-32. Let X1 and X2 denote the length and width of a part with standard deviations 0.1 and 0.2 centimeters, respectively. Suppose that the covariance
Weights of parts are normally distributed with variance σ2. Measurement error is normally distributed with mean 0 and variance 0.5σ2, independent of the part weights, and adds to the part weight.
An article in Knee Surgery Sports Traumatology, Arthroscopy [“Effect of Provider Volume on Resource Utilization for Surgical Procedures” (2005, Vol. 13, pp. 273–279)] showed a mean time of 129
Making handcrafted pottery generally takes two major steps: wheel throwing and fi ring. The time of wheel throwing and the time of firing are normally distributed random variables with means of 40
X and Y are independent, normal random variables with E(X) = 2, V (X) = 5, E(Y ) = 6, and V (Y ) = 8.Determine the following:(a) E(3X + 2Y ) (b) V (3X + 2Y )(c) P(3X + 2Y <18) (d) P(3X +
Suppose that X has a standard normal distribution. Let the conditional distribution of Y given X = x be normally distributed with mean E(Y | x) = 2x and variance V(Y | x) = 2x. Determine the
Patients given drug therapy either improve, remain the same, or degrade with probabilities 0.5, 0.4, 0.1, respectively. Suppose that 20 patients (assumed to be independent) are given the therapy. Let
In an acid-base titration, a base or acid is gradually added to the other until they have completely neutralized each other. Let X and Y denote the milliliters of acid and base needed for
Suppose that X and Y have a bivariate normal distribution with σX = 0.04, σY = 0.08, μX = 3.00, μY = 7.70, and ρ = 0.Determine the following:(a) P(2.95< X <3.05) (b) P(7.60(c)
Four electronic ovens that were dropped during shipment are inspected and classifi ed as containing either a major, a minor, or no defect. In the past, 60% of dropped ovens had a major defect, 30%
A Web site uses ads to route visitors to one of four landing pages. The probabilities for each landing page are equal. Consider 20 independent visitors and let the random variables W, X, Y , and Z
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
Test results from an electronic circuit board indicate that 50% of board failures are caused by assembly defects, 30% by electrical components, and 20% by mechanical defects. Suppose that 10 boards
Determine the covariance and correlation for the lengths of the minor and major axes in Exercise 5-29.Exercise 5-29The lengths of the minor and major axes are used to summarize dust particles that
Determine the covariance and correlation for the CD4 counts in a month and the following month in Exercise 5-30.Exercise 5-30An article in Health Economics [€œEstimation of the Transition
An article in Electronic Journal of Applied Statistical Analysis [“Survival Analysis of Acute Myocardial Infarction Patients Using Non-Parametric and Parametric Approaches” (2009, Vol. 2(1), pp.
An article in Electric Power Systems Research [“On the Self-Scheduling of a Power Producer in Uncertain Trading Environments” (2008, Vol. 78(3), pp. 311–317)] considered a self-scheduling
An article in Journal of Theoretical Biology [“Computer Model of Growth Cone Behavior and Neuronal Morphogenesis” (1995, Vol. 174(4), pp. 381–389)] developed a model for neuronal morphogenesis
Among homeowners in a metropolitan area, 25% recycle paper each week. A waste management company services 10,000 homeowners (assumed independent). Approximate the following probabilities:(a) More
Provide approximate sketches for beta probability density functions with the following parameters. Comment on any symmetries and show any peaks in the probability density functions in the
Consider the regional right ventricle transverse wall motion in patients with pulmonary hypertension (PH). The right ventricle ejection fraction (EF) is approximately normally distributed with
An article in IEEE Journal on Selected Areas in Communications [“Impulse Response Modeling of Indoor Radio Propagation Channels” (1993, Vol. 11(7), pp. 967–978)] indicated that the successful
Suppose that the construction of a solar power station is initiated. The project’s completion time has not been set due to uncertainties in financial resources. The proportion of completion within
The waiting time for service at a hospital emergency department follows an exponential distribution with a mean of three hours. Determine the following:(a) Waiting time is greater than four hours(b)
The size of silver particles in a photographic emulsion is known to have a log normal distribution with a mean of 0.001 mm and a standard deviation of 0.002 mm.(a) Determine the parameter values for
Suppose that the construction of a solar power station is initiated. The project’s completion time has not been set due to uncertainties in financial resources. The completion time for the first
An allele is an alternate form of a gene, and the proportion of alleles in a population is of interest in genetics. An article in BMC Genetics [“Calculating Expected DNA Remnants From Ancient
The maximum time to complete a task in a project is 2.5 days. Suppose that the completion time as a proportion of this maximum is a beta random variable with α = 2 and β = 3. What is the
The length of stay at a hospital emergency department is the sum of the waiting and service times. Let X denote the proportion of time spent waiting and assume a beta distribution with α = 10 and β
A European standard value for a low-emission window glazing uses 0.59 as the proportion of solar energy that enters a room. Suppose that the distribution of the proportion of solar energy that enters
Suppose that X has a beta distribution with parameters α = 1 and β = 4.2. Determine the following:(a) P(X < 0.25)(b) P(0.5 < X)(c) Mean and variance
Suppose that x has a beta distribution with parameters α = 2.5 and β = 1. Determine the following:(a) P(X < 0.25)(b) P(0.25 < X < 0.75)(c) Mean and variance
Suppose that X has a beta distribution with parameters α = 2.5 and β = 2.5. Sketch an approximate graph of the probability density function. Is the density symmetric?
Consider the lifetime of a laser in Example 4-26. Determine the following in parts (a) and (b):(a) Probability the lifetime is less than 1000 hours(b) Probability the lifetime is less than 11,000
An article in Chemosphere [“Statistical Evaluations Reflecting the Skewness in the Distribution of TCDD Levels in Human Adipose Tissue” (1987, Vol.16(8), pp. 2135- 2140)] concluded that the
An article in Applied Mathematics and Computation [“Confidence Intervals for Steady State Availability of a System with Exponential Operating Time and Lognormal Repair Time” (2003, Vol.137(2),
An article in Journal of Hydrology [“Use of a Lognormal Distribution Model for Estimating Soil Water Retention Curves from Particle-Size Distribution Data” (2006, Vol. 323(1), pp. 325–334)]
Suppose that the length of stay (in hours) at a hospital emergency department is modeled with a lognormal random variable X with θ = 1.5 and ω = 0.4. Determine the following in parts (a) and
An article in Health and Population: Perspectives and Issues (2000, Vol. 23, pp. 28–36) used the lognormal distribution to model blood pressure in humans. The mean systolic blood pressure (SBP) in
An article in IEEE Transactions on Dielectrics and Electrical Insulation [“Statistical Analysis of the AC Breakdown Voltages of Ester Based Transformer Oils” (2008, Vol. 15(4))] used Weibull
An article in Financial Markets Institutions and Instruments [“Pricing Reinsurance Contracts on FDIC Losses” (2008, Vol. 17(3)] modeled average annual losses (in billions of dollars) of the
An article in Electronic Journal of Applied Statistical Analysis [“Survival Analysis of Dialysis Patients Under Parametric and Non-Parametric Approaches” (2012, Vol. 5(2), pp. 271–288)] modeled
An article in the Journal of the Indian Geophysical Union titled “Weibull and Gamma Distributions for Wave Parameter Predictions” (2005, Vol. 9, pp. 55–64) described the use of the Weibull
An article in Mathematical Biosciences [“Influence of Delayed Viral Production on Viral Dynamics in HIV-1 Infected Patients” (1998, Vol.152(2), pp. 143–163)] considered the time delay between
An article in Sensors and Actuators A: Physical [“Characterization and Simulation of Avalanche PhotoDiodes for Next-Generation Colliders” (2011, Vol.172(1), pp.181– 188)] considered an
The total service time of a multi step manufacturing operation has a gamma distribution with mean 18 minutes and standard deviation 6.(a) Determine the parameters λ and r of the distribution.(b)
Patients arrive at a hospital emergency department according to a Poisson process with a mean of 6.5 per hour.(a) What is the mean time until the 10th arrival?(b) What is the probability that more
Given the probability density function f(x) = 0.013 x2e−0.01x / Γ(3), determine the mean and variance of the distribution.
An article in Vaccine [“Modeling the Effects of Influenza Vaccination of Health Care Workers in Hospital Departments” (2009, Vol.27(44), pp. 6261–6267)] considered the immunization of
Requests for service in a queuing model follow a Poisson distribution with a mean of five per unit time.(a) What is the probability that the time until the first request is less than 4 minutes?(b)
An article in Journal of National Cancer Institute [“Breast Cancer Screening Policies in Developing Countries: A Cost-Effectiveness Analysis for India” (2008, Vol.100(18), pp. 1290–1300)]
The length of stay at a specific emergency department in a hospital in Phoenix, Arizona had a mean of 4.6 hours. Assume that the length of stay is exponentially distributed.(a) What is the standard

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