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applied statistics and probability for engineers

Questions and Answers of Applied Statistics And Probability For Engineers

Prove that for any random variables X and Y with finite variances:(a) cov(X,Y ) Æ cov(X,E[Y j X]).(b) X and Y ¡E[Y j X] are uncorrelated.
Suppose that the distribution of Y conditional on X Æ x is N(x,x2) and the marginal distribution of X isU[0,1].(a) Find E[Y ] .(b) Find var[Y ] .
Let X1 » gamma(r,1) and X2 » gamma(s,1) be independent. Find the distribution of Y ÆX1 ÅX2.
Suppose that X and Y are jointly normal, i.e. they have the joint PDF:(a) Derive the marginal distribution of X and Y and observe that both are normal distributions.(b) Derive the conditional
Prove the following: If (X1,X2, . . . ,Xm) are pairwise uncorrelated m m var X var [xi]. i=1 i=1
Let X and Y be independent random variables withmeans ¹X , ¹Y , and variances ¾2 X , ¾2 Y .Find an expression for the correlation of XY and Y in terms of thesemeans and variances.Hint: “XY ”
Show that any random variable is uncorrelated with a constant.
Let X and Y have density f (x, y) Æ 12x y(1¡ y) for 0 Ç x Ç 1 and 0 Ç y Ç 1. Are X and Y independent or dependent?
Let X and Y have density f (x, y) Æ 1 on 0 Ç x Ç 1 and 0 Ç y Ç 1. Find the density function of Z Æ XY .
Let X and Y have density f (x, y) Æ exp(¡x¡y) for x È 0 and y È 0. Find the marginal density of X and Y . Are X and Y independent or dependent?
Let the joint PDf of X and Y be given by(a) Find the value of c such that f (x, y) is a joint PDF.(b) Find themarginal distributions of X and Y .(c) Are X and Y independent? Compare your answer to
Let F(x, y) be the distribution function of (X,Y ). Show that P[a Ç X · b,c Ç Y · d] Æ F (b,d)¡F (b,c)¡F (a,d)ÅF (a,c) .
Let the joint PDF of X and Y be given by f (x, y) Æ g (x)h(y) for some functions g (x) and h(y). Let a ÆR 1¡1 g (x)dx and b ÆR 1¡1h(x)dx.(a) What conditions a and b should satisfy in order for f
Letfor 0 · x and 0 · y.(a) Verify that f (x, y) is a valid density function.(b) Find themarginal density of X.(c) Find E[Y ] , var[Y ], E[XY ] and corr(X,Y ).(d) Find the conditional density of Y
Let f (x, y) Æ x Å y for 0 · x · 1 and 0 · y · 1 (and zero elsewhere).(a) Verify that f (x, y) is a valid density function.(b) Find themarginal density of X.(c) Find E[Y ] , var[X], E[XY ] and
Let f (x, y) Æ 1/4 for ¡1 · x · 1 and ¡1 · y · 1 (and zero elsewhere).(a) Verify that f (x, y) is a valid density function.(b) Find themarginal density of X.(c) Find the conditional density of
For the mixture of normals distribution show(a)R 1¡1 f (x)dx Æ 1.(b) F (x) ÆPM mÆ1 pm©µx ¡¹m¾m¶.(c) E[X] ÆPM mÆ1 pm¹m.(d) E£X2¤ÆPM mÆ1 pm¡¾2 m Å¹2 m¢.
For the lognormal distribution show (a) The density is obtained by the transformation X Æ exp(Y ) with Y » N(µ, v).(b) E[X] Æ exp(µÅv/2).
For the logistic distribution show(a) F(x) is a valid distribution function.(b) The density function is f (x) Æ exp(¡x)/¡1Åexp(¡x)¢2Æ F(x) (1¡F(x)).(c) The density f (x) is symmetric about
For the Pareto distribution show(a)R 1¯ f¡x j ®,¯¢dx Æ 1.(b) F¡x j ®,¯¢Æ 1¡¯®x® , x ¸ ¯.(c) E[X] Æ®¯®¡1.(d) var[X] Æ®¯2(®¡1)2 (®¡2).
Suppose X » gamma(®,¯). Set Y Æ ¸X. Find the density of Y . Which distribution is this?
For the gamma distribution show(a)R 1 0 f¡x j ®,¯¢dx Æ 1.(b) E[X] Æ®¯.(c) var[X] Æ®¯2 .
Show Theorem 3.3. Hint: Show that x¡1 f (x j r ) Æ 1 r¡2 f (x j r ¡2).
For the chi-square density f (x j r ) show(a)R 1 0 f (x j r )dx Æ 1.(b) E[X] Æ r.(c) var[X] Æ 2r .
For the double exponential distribution show(a)R 1¡1 f (x j ¸)dx Æ 1.(b) E[X] Æ 0.(c) var[X] Æ 2¸2.
For the exponential distribution show(a)R 1 0 f (x j ¸)dx Æ 1.(b) E[X] Æ ¸.(c) var[X] Æ ¸2.
For theU[a,b] distribution show(a)R b a f (x j a,b)dx Æ 1.(b) E[X] Æ (b ¡a)/2(c) var[X] Æ (b ¡a)/12.
For the Poisson distribution show(a)1X xÆ0¼(x j ¸) Æ 1.(b) E[X] Æ ¸.(c) var[X] Æ ¸.
For the Binomial distribution show(a)Xn xÆ0¼¡x j n,p¢Æ 1.Hint: Use the Binomial Theorem.(b) E[X] Æ np.(c) var[X] Æ np(1¡p).
For the Bernoulli distribution show(a)X1 xÆ0¼¡x j p¢Æ 1.(b) E[X] Æ p.(c) var[X] Æ p(1¡p).
First-Order Stochastic Dominance. A distribution F(x) is said to first-order dominate distributionG(x) if F(x) ·G(x) for all x and F(x) ÇG(x) for at least one x. Showthe following proposition:F
Surveys routinely ask discrete questions when the underlying variable is continuous. For example, wage may be continuous but the survey questions are categorical. Take the following example.wage
Let X have density f (x) Æ e¡x for x ¸ 0. Suppose X is censored to satisfy X¤ ¸ c È 0. Find the mean of the censored distribution.
Let X » U[0,1] be uniformly distributed on [0, 1]. (X has density f (x) Æ 1 on [0, 1], zero elsewhere.) Suppose X is truncated to satisfy X · c for some 0 Ç c Ç 1.(a) Find the density function
Suppose the random variable X is a duration (a period of time). Examples include: a spell of unemployment, length of time on a job, length of a labor strike, length of a recession, length of an
Let X be a random variable with mean ¹ and variance ¾2. Show that E h¡X ¡¹¢4 i¸ ¾4.
Let X be a random variable with E[X] Æ 1. Show that E£X2¤È 1 if X is not degenerate.Hint: Use Jensen’s inequality.
The skewness of a distribution is skew Æ¹3¾3 where ¹3 is the 3rd central moment.(a) Show that if the density function is symmetric about some pointa, then skew Æ 0.(b) Calculate skew for f (x)
Show that if X is a continuous random variable, then min aEjX ¡aj Æ EjX ¡mj , wherem is the median of X.Hint: Work out the integral expression of EjX ¡aj and notice that it is differentiable.
Find a which minimizes E£(X ¡a)2¤. Your answer should be amoment of X.
Find the median of the density f (x) Æ 12 exp(¡jxj) , x 2 R.
Suppose X has density f (x) Æ e¡x on x È 0. Set Y Æ ¡logX. Find the density of Y.
Suppose X has density f (x) Æ ¸¡1e¡x/¸ on x È 0 for some ¸ È 0. Set Y Æ X1/® for ® È 0.Find the density of Y .
Suppose X has density f (x) Æ e¡x on x È 0. Set Y Æ ¸X for ¸ È 0. Find the density of Y .
Show that if the density satisfies f (x) Æ f (¡x) for all x 2 R then the distribution function satisfies F(¡x) Æ 1¡F(x).
Let X have density fX (x) Æ1 2r /2¡¡ r 2¢ xr /2¡1 exp³¡x 2´for x ¸ 0. This is known as the chi-square distribution. Let Y Æ 1/X. Show that the density of Y is fY (y) Æ1 2r /2¡¡ r 2¢
Compute E[X] and var[X] for the following distributions.(a) f (x) Æ ax¡a¡1, 0 Ç x Ç 1, a È 0.(b) f (x) Æ 1n, x Æ 1, 2, ...,n.(c) f (x) Æ 32(x ¡1)2, 0 Ç x Ç 2.
Find the mean and variance of X with density f (x) Æ1¸exp³¡x¸´.
A Bernoulli random variable takes the value 1 with probability p and 0with probability 1¡p X Æ½1 with probability p 0 with probability 1¡p.Find the mean and variance of X.
Define F(x) Æ½0 if x Ç 0 1¡exp(¡x) if x ¸ 0.(a) Show that F(x) is a CDF.(b) Find the PDF f (x).(c) Find E[X] .(d) Find the PDF of Y Æ X1/2.
Let X »U[0,1]. Find the distribution function of Y Æ logµX 1¡X¶.
Let X »U[0,1]. Find the PDF of Y Æ X2.
In the poker game “Five Card Draw" a player first receives five cards drawn at random.The player decides to discard some of their cards and then receives replacement cards. Assume a player is dealt
Consider drawing five cards at random from a standard deck of playing cards. Calculate the following probabilities.(a) A straight (five cards in sequence, suit not relevant).(b) A flush (five cards
In the game of blackjack you are dealt two cards from a standard playing deck. Your score is the sum of the value of the two cards, where numbered cards have the value given by their number, face
Monte Hall. This is a famous (and surprisingly difficult) problem based on an old U.S.television game show “Let’s Make a Deal hosted by Monte Hall”. A standard part of the show ran as follows:
Sometimes we use the concept of conditional independence. The definition is as follows:let A,B,C be three events with positive probabilities. Then A and B are conditionally independent given C if
Suppose 1% of athletes use banned steroids. Suppose a drug test has a detection rate of 40% and a false positive rate of 1%. If an athlete tests positive what is the conditional probability that the
Suppose that the unconditional probability of a disease is 0.0025. A screening test for this disease has a detection rate of 0.9, and has a false positive rate of 0.01. Given that the screening test
If four random cards are dealt from a deck of playing cards, what is the probability that all four are Aces?
Calculate the following probabilities, assuming fair coins and dice.(a) Getting three heads in a row fromthree coin flips.(b) Getting a heads given that the previous coin was a tails.(c) From two
You are on a game show, and the host shows you five doors marked A, B, C, D, and E. The host says that a prize is behind one of the doors, and you win the prize if you select the correct door.Given
Calculate the following probabilities concerning a standard 52-card playing deck.(a) Drawing a King with one card.(b) Drawing a King on the second card, conditional on a King on the first card.(c)
Give an example where P[A] È 0 yet P[A j B] Æ 0.
Is P[A j B] · P[A], P[A j B] ¸ P[A] or is neither necessarily true?
Prove that P[A\B \C] Æ P[A j B \C]P[B jC]P[C] .Assume P[C] È 0 and P[B \C] È 0.
Suppose A\B Æ A. Can A and B be independent? If so, give the appropriate condition.
Show that P[A\B] · P[A] · P[A[B] · P[A]ÅP[B].
Prove that P[A[B] Æ P[A]ÅP[B]¡P[A\B].
If P[A] Æ 1/2 and P[B] Æ 2/3, can A and B be disjoint? Explain.
For events A and B, express “either A or B but not both” as a formula in terms of P[A], P[B], and P[A\B].
Froma 52-card deck of playing cards draw five cards tomake a hand.(a) Let A be the event “The hand has two Kings”. Describe Ac .(b) A straight is five cards in sequence, e.g. {5,6,7,8,9}. A flush
Describe the sample space S for the following experiments.(a) Flip a coin.(b) Roll a six-sided die.(c) Roll two six-sided dice.(d) Shoot six free throws (in basketball).
Let A Æ {a,b,c,d} and B Æ {a,c,e, f }.(a) Find A\B.(b) Find A[B.
=+d. Considering the number of prior nonsoccer concussions, the values of mean 6 sd for the threegroups were .30 6 .67, .49 6 .87, and .19 6 .48.Analyze this data and draw appropriate conclusions.
=+c. Here is summary information on score on a controlled oral word association test for the soccer and nonsoccer athletes:n1 5 26 x15 37.50 s1 5 9.13 n2 5 56 x25 39.63 s2 5 10.19 Analyze this data
=+b. For the soccer players, the sample correlation coefficient calculated from the values of x 5 soccer exposure (total number of competitive seasons played prior to enrollment in the study)and y 5
=+a. The paper reported that 45 of the 91 soccer players in their sample had suffered at least one concussion, 28 of 96 nonsoccer athletes had suffered at least one concussion, and only 8 of 53
=+72. Have you ever wondered whether soccer players suffer adverse effects from hitting “headers”? The authors of the article “No Evidence of Impaired Neurocognitive Performance in Collegiate
=+ratio as the dependent variable. Use a statistical software package to fit several different regression models, and draw appropriate inferences.
=+b. The investigators fit the simple linear regression model to the entire data set consisting of 30 observations, with ppv as the independent variable and
=+a. Construct a comparative boxplot of ppv for the cracked and uncracked prisms, and comment.Then estimate the difference between true average ppv for cracked and uncracked prisms in a way that
=+71. Curing concrete is known to be vulnerable to shock vibrations, which may cause cracking or hidden damage to the material. As part of a study of vibration phenomena, the paper “Shock
=+a. Use various techniques to decide whether it is plausible that the two techniques measure on average the same amount of fat.
=+70. The article “Evaluating the BOD POD for Assessing Body Fat in Collegiate Football Players”(Medicine and Science in Sports and Exercise, 1999: 1350–1356) reports on a new air
=+to a stress protocol and then removed and tested at various times after the protocol had been applied.The accompanying data on x 5 time (min) and y 5 blood glucose level (mmol/L) was read from a
=+69. Normal hatchery processes in aquaculture inevitably produce stress in fish, which may negatively impact growth, reproduction, flesh quality, and susceptibility to disease. Such stress
=+« denotes a random deviation and , , and are parameters. Estimate the model parameters, and obtain a prediction interval for wear life when speed is 60 rpm and load is 6000 psi.(Hint:
=+b. The cited article contains the comment that a lognormal distribution is appropriate for wear life, since ln(w) is known to follow a normal law.The suggested model is w 5 3y(sl )4«, where
=+a. With w 5 wear life, s 5 speed, and l 5 load (in 1000s), fit the model with dependent variable w and predictors s and l, and assess the utility of the fitted model.
=+68. Three sets of journal bearing tests were run on a Mil-L-8937-type film at each combination of three loads (psi) and three speeds (rpm). The wear life(hr) was recorded for each run, resulting in
=+b. Express the estimated regression in uncoded form.c. SSTo 5 17.2567 and R2 for the model of part (a)is .885. When a model that includes only the four independent variables as predictors is fit,
=+Calculate a point prediction of brightness when H2O2 is .4%, NaOH is .4%, silicate is 3.5%, and temperature is 175. What are the values of the residuals for the observations made with these values
=+were .650, 2.258, .133, and .108, respectively; the estimated quadratic coefficients were 2.135, .028,.028, and –.072, respectively; and the estimated coefficients of the interaction predictors
=+a. When the complete second-order coded model was fit, the estimate of the constant term was 84.67;the estimated coefficients of the linear predictors
=+analysis as follows:Coded Variable value: –2 –1 0 1 2 H2O2 .1 .2 .3 .4 .5 NaOH .1 .2 .3 .4 .5 Silicate .5 1.5 2.5 3.5 4.5 Temperature 130 145 160 175 190 The data follow:Obs H2O2 NaOH Silicate
=+67. A study was carried out to investigate the relationship between brightness of finished paper (y) and the variables percentage of H2O2 by weight, percentage of NaOH by weight, percentage of
=+ predictors and six xi xj predictors). Use a statistical computer package to identify a good model based on this candidate pool of predictors.
=+Obs Formaldehyde concentration Catalyst ratio Curing temperature Curing time Durable press rating 6 7 7 180 1 4.7 7 7 13 140 1 4.6 8 5 4 160 7 4.5 9 4 7 140 3 4.8 10 5 1 100 7 1.4 11 8 10 140 3 4.7
=+c. Given that catalyst ratio, curing temperature, and curing time all remain in the model, do you think that formaldehyde concentration provides useful information about durable press rating?Data

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