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nonparametric statistical inference

Probability And Statistical Inference 3rd Edition Robert Bartoszynski, Magdalena Niewiadomska-Bugaj - Solutions

An experiment consists of tossing a fair coin 13 times. Such an experiment is repeated 17 times. Find the probability that in a majority of repetitions of the experiment the tails will be in minority.
Assume that X1 and X2 are independent random variables, with BIN(n1,p) and BIN(n2,p) distributions, respectively. Find a correlation coefficient between X1 and X1 + X2.
Suppose that random variables X1, ...,Xn are independent, each with the same Bernoulli distribution. Given that n i=1 Xi = r, find: (i) The probability that X1 = 1.(ii) The covariance between Xi and Xj , 1 ≤ i
Assume that we score Y = 1 for a success and Y = −1 for a failure. Express Y as a function of the number X of successes in a single Bernoulli trial, and find moments E(Y n), n =1, 2, . . . .
Label statements below as true or false:(i) Suppose that 6% of all cars in a given city are Toyotas. Then the probability that there are 4 Toyotas in a row of 12 cars parked in the municipal parking is 12 4(0.06)4(0.94)8. (ii) Suppose that 6% of all Europeans are French. Then the probability that
Show that if X has the Poisson distribution with mean λ, then P(X ≤ λ/2) ≤ (2/e)λ/2 and P(X ≥ 2λ) ≤ (e/4)λ. (Hint: Use the inequality in Problem 7.8.7, and find minimum of the right-hand sides for t.)
Let X be a random variable such that a mgf mX(t) exists for all t. Use the same argument as in the proof of the Chebyshev inequality to show that P{X ≥ y} ≤e−tymX(t),t ≥ 0.
Let X have the Poisson distribution with mean λ. Show that PX ≤λ2≤4λ and P{X ≥ 2λ} ≤ 1λ
Show that if X has a mgf bounded by the mgf of exponential distribution (i.e., λ/(λ −t) for t 1 we have P{X>} ≤ λe−(λ−1).(Hint: Use the mgf of exponential distribution to obtain the bound for P{X>}, then determine its minimum.)
Derive the Chebyshev inequality from the Markov inequality.
Prove the Markov inequality when X is a continuous random variable.
Assume that E(X) = 12,P(X ≥ 14) = 0.12, and P(X ≤ 10) = 0.18. Show that Var(X) is at least 1.2.
Let X be any random variable. Show that E(X) ≥ 0.2 if P(X ≥ 0) = 1 and P(X ≥2) = 0.1.
Let X1, ...,Xn be independent variables with a U[0, 1] distribution. The joint density of the S = min(X1, ...,Xn) and T = max(X1, ...,Xn) is f(s,t) = n(n − 1)(t − s)n−2 for 0 ≤ s ≤ t ≤ 1 and f(s,t)=0 otherwise. Find:(i) E(Sm),E(T m), and ρ(S,T). (ii) E(T|S).
The number of traffic accidents that occur in a certain city in a week is a random variable with mean μ and variance σ2. The numbers of people injured in an accident are independent random variables, each with mean m and variance k2. Find the mean and variance of the number of people injured in a
Let X,Y be continuous random variables with a joint density f(x,y). Assume that E(Y |X = x) = μ for all x. Show that Var(Y ) =Var(Y |X = x)fX(x) dx.
Let X and Y have the joint density uniform on the triangle with vertices (0, 0),(2, 0)and (3, 1). Find: (i) E(X|Y ) and E(Y |X). (ii) Var(X|Y ) and Var(Y |X). (iii) The expectations and variances of X and Y using formulas (7.47) and (7.48).
Variables X and Y are jointly distributed with the density f(x,y) = Cx(3x + 2y) for 0
Let X1, ...Xn be independent random variables having the same distribution with a mean μ and variance σ2. Let X = (X1 + ··· + Xn)/n. Show that E{n i=1 (Xi − X)2} = (n − 1)σ2. [Hint: Since Xi − X = (Xi − μ) − (X − μ), we have n i=1 (Xi − X)2 = n i=1 (Xi − μ)2 − n(X −
Let X,Y be independent, with means μX,μY and variances σ2 X,σ2 Y . Show that Var(XY ) = σ2 Xσ2 Y + σ2 Xμ2 Y + σ2 Y μ2 X.
Find the correlation of random variables X and Y jointly uniformly distributed on:(i) The triangle with vertices (0, 0),(1, 0),(1, 2). (ii) The quadrangle with vertices(0, 0),(a, 0),(a, 2), and (2a, 2), where a > 0.
Let variables X and Y be such that E(X) = E(Y )=0, Var(X) = Var(Y )=1, andρX,Y = ρ. Find E(W), Var(W), and ρW,Y if W = X − ρY .
Random variables X and Y are jointly distributed with the density f(x,y) =(24/11)y(1 − x − y) for x > 0,y> 0, and x + y < 1. Find the Cov(X,Y ).
Let mX(t) = p + 0.5e−t + (0.5 − p)et be a mgf of a variable X. Determine possible values of p and find Var(X27) as a function of p.
Find the variance of a random variable X with a cdf F(x) =⎧⎪⎨⎪⎩0 for x < 0√x for 0 ≤ x ≤ 1 1 for x > 1.
For random variables X and Y such that E(X)=2,E(Y )=1,E(X2) =10,E(Y 2)=3, and E(XY ) =c, find: (i) Var(3X − 5Y ). (ii) ρX,Y . (iii) The range of values of c for which the assumptions of the problem are consistent.
Find the variance of variable X if its first ordinary moment is 3 and the second factorial moment is 52.
A random variable X has binomial distribution with mean 5 and standard deviation 2. Find P{X = 6}.
Show that the distribution of X is symmetric around 0 if and only if ϕX(t) is real.
Show that if ϕ(t) is a chf, then |ϕ(t)|2 is also a chf.
A family G of distributions is said to be closed under convolution, if whenever independent random variables X and Y have distributions in G; the same is true for the random variable X + Y . Show closeness under convolution in families of: (i) Poisson distributions; (ii) Normal distributions.
Find the chf of the following distributions: (i) POI(λ). (ii) GEO(p). (iii) EXP(λ). (iv)N(μ,σ2).
Let random variable X have mgf mX(t)=0.2+0.4e−t + 0.1et + 0.3e2t. Use mX to obtain E[X(X − 1)(X − 2)]—the third factorial moment of X.
Let X be a random variable with E(X) = μ,E(X − μ)2 = σ2 and such that γ4 =E(X − μ)4 exists. Then γ4/σ4 is called the coefficient of kurtosis. Find kurtosis of the following distributions: (i) N(0, 1). (ii) N(μ,σ2). (iii) BIN(1,p). (iv) POI(λ).
Let X be a random variable with E(X) = μ,E(X − μ)2 = σ2 and such that the third central moment γ3 = E(X − μ)3 exists. The ratio γ3/σ3 is called the coefficient of skewness. Find skewness of the following distributions: (i) U[0, 1]. (ii) f(x) = αxα−1 for 0 ≤ x ≤ 1 and f(x)=0
A continuous random variable X is called symmetric about c if its density f satisfies the condition f(c − x) = f(c + x) for all x. Show that: (i) If X is symmetric about c and E(X) exists, then E(X) =c. (ii) If X is symmetric about 0, then all moments of odd order (if they exist) are equal to 0.
Let X1, ...,Xn be independent random variables with the same distribution N(μ,σ2). Find constants αn and βn such that U = (X1 + ··· + Xn − αn)/βn and X1 have the same distribution.
Find the mgf for a random variable with a density: (i) f(x) = xe−x for x > 0. (ii)f(x; σ2) = 2/πe−x2/2σ2 for x > 0.
Find the fourth factorial moment of the random variable X with a POI(λ) distribution.
Let X be a nonnegative integer-valued random variable. The function gX(s) = EsX, defined for |s| ≤ 1, is called a probability generating function, or simply a generating function, of X. Find gX(s) for random variables with: (i) Geometric distribution.(ii) Binomial distribution. (iii) Poisson
Find the mgf of a discrete random variable X with distribution P{X = k} =1/n,k = 0, 1, ...,n − 1.
We say that X isstochastically smallerthan Y (X ≤stY ) if P{X ≤ t} ≥ P{Y ≤ t} for all t. Show that if X and Y have finite expectations and X≤stY , then E(X) ≤ E(Y ).(Hint: Start with nonnegative X and Y and use Theorem 7.2.2. Then use the decomposition into a positive and negative
Show that if X is such that P(a ≤ X ≤ b)=1, then E(X) exists and a ≤ E(X) ≤ b.
A cereal company puts a plastic bear in each box of cereal. Every fifth bear is red.If you have three red bears, you get a free box of the cereal. If you decide to keep buying this cereal until you get one box free, how many boxes would you expect to buy before getting a free one? [Hint: Represent
An urn contains w white and r red balls. We draw n ≤ r balls from the urn without replacement, and we let X be the number of red balls drawn. Find: E(X), by defining indicator variables: (i) X1, ...,Xn such that X = X1 + ··· + Xn. (ii) Y1, ··· ,Yr such that X = Y1 + ··· + Yr.
Let X have the density f(x) =⎧⎪⎨⎪⎩cx 0 ≤ x ≤ 1 c(2 − x) 1 ≤ x ≤ 2 0 otherwise.Find: (i)c. (ii) E(X). (iii) E(2 − X)3. (iv) E[1/(2 − X)].
Let X be a random variable with density f(x)=1/2 for −1 ≤ x ≤ 1 and f(x)=0 otherwise. Find: (i) E(X). (ii) E(X2). (iii) E(2X − 3)2.
Find E(X),E(1/X), and E(2X) if X = 1, 2, ..., 8 with equal probabilities 1/8.
Assume that X has density f(x) = ax + bx3 for 0 ≤ x ≤ 2 and f(x)=0 otherwise.Find a and b if E(X2)=2.5.
Show that E(X − E(X)) = 0.
Show that if the expectation of a continuous type random variable X exists and the density f(x) of X satisfies the condition f(x) = f(2a − x) for all x ≥ 0, then E(X) = a.
A point is randomly selected from the interval (0, 1). Find the expected length of the smaller part of the interval.
Random variables X and Y are jointly distributed with density f(x,y) = 12(x − y)2 for 0 ≤ x
Let X be a nonnegative continuous random variable with hazard rate h(t) = t. Find E(X).
Let X1,X2, ...,Xn be independent, each with the same distribution. Find E[min(X1, ...,Xn)] if the distribution of variables is (i) EXP(λ). (ii) U[a,b]. [Hint:First find the cdf (or density) of the random variable whose expectation is being computed.]
Find the expected value E(|X|), where X is a normal random variable with parameters μ = 0 and σ2 (the distribution of |X| is called folded normal or half normal).
Suppose there are k = 10 types of toys (plastic animals, etc.) to be found in boxes of some cereal. Assume that there is a toy in every box and that each type of toy occurs with equal frequency. (i) Find E(X), where X is the number of boxes you must buy until you collect three different types of
The density of the lifetime T of some part of electronic equipment is f(t) = λ2te−λt,t> 0. Find E(T).
Variable X assumes values 0, 1, 2, and 3 with probabilities 0.3,a, 0.1, andb, respectively. Find a and b if: (i) E(X) = 1.5. (ii) E(X) = m. First determine all possible values of m.
Let four observations X1, X2, X3, X4 be independently selected from the same distribution with density f(x) = e−x for x > 0 and 0 otherwise. Find: (i) The probability that exactly one of these observations is less than 1. (ii) P(X1 + X2 < X3 + X4).
A can of Three-Bean-Salad contains beans of varieties A, B, and C (plus other ingredients which are of no concern for the problem). Let X, Y, and Z denote the relative weights of varieties A, B, and C in a randomly selected can (so that X + Y + Z = 1).Moreover, let the joint distribution of (X, Y )
A lifetime of some electronic unit has density f(t). Each time the unit fails, it is replaced by another one with the same lifetime distribution. Let X, Y, and Z be lifetimes of three consecutive units independently installed one after the other failed.Find: (i) P(X
Assume that variables X, Y, and Z are jointly distributed with the density f(x, y, z) = x + y2 + z5 for 0
Let X, Y , and Z have joint density f(x, y, z) = c(x + y + z) for 0
Suppose that X1, X2, X3 have joint density f(x1, x2, x3) = x1x2x3 for 0 < xi
Two cards are drawn without replacement from an ordinary deck. Let X be the number of aces, Y be the number of red cards, and Z be the number of hearts. Find: (i) The joint distribution of (X, Y, Z). (ii) P{Y = Z}. (iii) The conditional distribution of Z given Y = y.
Let variables X1, X2, and X3 be independent, each having a U(0, 1) distribution.Find: (i) The probability that exactly two of the three variables will be larger than 0.4.(ii) P(X1 + X2 > X3).
Let variables X1 and X2 be independent, such that P(Xi = −1) = P(Xi = 1) = 0.5, i = 1, 2. Moreover, let X3 = X1X2. Show that variables X1, X2, X3 are not independent but each two of them are independent (they are pairwise independent).
A current of I amperes following through a resistance of R ohms varies according to the probability distribution with density f(i) = 6i(1 − i) 0
Darts are thrown at a circular target. Let variables X and Y , coordinates of the point of impact be independent each having N(0, σ2) distribution. Find the density of D, the distance of the point of impact from the center of the target.
Let f be the joint density of a pair (X, Y ) of random variables, and let a and b be two constants. Find the densities of (i) aX + bY . (ii) XY . (iii) X/Y . (iv) (U, V ), where U = aX +b, V = cY +d, and ac = 0 .
Let R be a nonnegative random variable with density f(r). Let (X, Y ) be a bivariate distribution obtained as follows: First, randomly choose a value of R, and then chose a value of U according to its U(0, 1) distribution. Now, put X = R cos(2πU), Y = R sin(2πU). (6.28)Find: (i) The joint density
The joint distribution of random variables X and Y has density f(x, y) = cx2,c> 0, for 0 0 and 0 otherwise. Find the distribution of W = X + Y .
Random variables X and Y have joint density f(x, y) = cx, c > 0, for −x < y < x, 0
Random variables (X, Y ) have a joint density f(x, y) = k(ax + by) if 0 0 are given constants. Find: (i) The value of k as a function of a andb. (ii) The density of variable Z = 1/(Y + 1)2. (Hint: Express F(t) = P{Z ≤ t} in terms of the cdf of Y .)
Independent random variables X and Y are uniformly distributed over intervals(−1, 1), and (0, 1), respectively. Find the joint distribution of variables U = XY and V = Y . Determine the support of the density function.
Let X and Y be independent random variables with densities fX(x) = c1xα−1e−x, fY (y) = c2yβ−1e−y for x > 0,y > 0,α> 0,β > 0, and normalizing constants c1, c2. Find the density of W = X/(X + Y ).
Let random variables X1, X2 be independent and have both EXP(λ) distribution.(i) Find the cdf and the density of X1 − X2. (ii) Find the cdf and the density of X1/X2. (iii) Show that Z1 = X1 + X2 and Z2 = X1/(X1 + X2) are independent.
Let X, Y be independent, each with a standard normal distribution. Find the distribution of: (i) V = X/Y . (ii) U = (X − Y )2/2. (iii) W = X/|Y |.
A fast-food restaurant has a dining room and a drive-thru window. Let X and Y be the fractions of time (during a working day) when the dining room (X) and the drive-thru window (Y ) are busy. Assuming that the joint density of (X, Y ) is f(x, y) = k(2x2 +y2) for 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 and
The phrase “A stick is broken at random into three pieces” can be interpreted in several ways. Let us identify the stick with interval [0, 1] and let 0
Let X and Y have joint density of the form f(x, y) = A(y − x)α for 0 ≤ x
Let variables X and Y be independent, each with U[0, 1] distribution. Find: (i) P(X +Y ≤ 0.5 | X = 0.25). (ii) P(X + Y ≤ 0.5 | X ≥ 0.25). (iii) P(X ≥ Y | Y ≥ 0.5).
Refer to Problem 6.2.15 and find: (i) The conditional density of X given Y and of Y given X. (ii) Probability that a randomly selected customer spent more than 30 minutes at the counter if he was served less than 15 minutes. (iii) Probability that customer’s waiting time was less than 10 minutes
Let X and Y have the joint density f(x, y) = λ2e−λy for 0 ≤ x ≤ y and f(x, y)=0 otherwise. Find: (i) The joint cdf of (X, Y ). (ii) The marginal densities of X and Y .(iii) The conditional density of Y given X.
Suppose that three cards are drawn without replacement from an ordinary deck. Let X be the number of aces among the cards drawn and Y be the number of red cards among them. Find: (i) The joint distribution of (X, Y ). (ii) The conditional distribution of the number of aces if it is known that all
Let X and Y have distribution uniform in the shape of the letter Y (see Figure 6.8).Identify the shapes of the marginal densities of X and Y in Figure 6.9.
Let X, Y be independent, continuous random variables with a symmetric (but possibly different) distribution around 0. Show that Y /X and Y /|X| have the same distribution. (Hint: Compare the cdf’s of W = X/Y and V = X/|Y |.)
Assume that X and Y are independent random variables with EXP(a) and EXP(b)distributions, respectively. Assume that it is not possible to observe both X and Y but that one can observe U = min(X, Y ) and Z =1 if X
Students have to randomly select a point inside a circular region with radius R. One of them first samples the direction from the center of the region according to a uniform distribution on [0◦, 360◦], and then samples the distance from the center according to U[0, R]. Find: (i) The density
Let X and Y be the time (in hours) that a customer spends at a service counter and the time he is actually being served (Y
Random variables X and Y have joint density f(x, y) = k(ax + by) 0 0. Find: (i) k (as a function of a and b). (ii) The marginal distributions of X and Y . (iii) The cdf of (X, Y ).
Let X and Y be the lifetimes of two components of a machine. Their joint distribution is given by the density f(x, y) = xe−x(1+y) for x > 0, y> 0 and zero, otherwise.(i) Find P(X ≥ 5). (ii) Find the probability that max(X, Y ) > 2. (iii) Check the independence of X and Y using their marginal
Let variables X and Y be independent, with distributions GAM(1, 1) and GAM(2, 1), respectively. Find: (i) P(X − Y > 0). (ii) P(|X − Y | < 1).
Let T1, T2 be independent random variables with hazard functions h1(t) and h2(t), respectively. (i) Show that the variable with the hazard function h(t) = h1(t) + h2(t)has the same distribution as min(T1, T2). (ii) Express P(T1 < T2) through h1 and h2.
Let (X, Y ) have the distribution given by the table
Let variables X and Y have joint density f(x, y) = 3(x + y)for x > 0,y > 0, x + y 0.5|X < 0.5).
Consider a system consisting of three components connected as in Figure 6.7. Let Y1, Y2, Y3 be independent lifetimes of components 1, 2, and 3, respectively, each with EXP (α) distribution. If T is the lifetime of the whole system, find: (i) The cdf of T.(ii) The hazard function of T.
Random variables have joint distribution given by the table X/Y 12 3 1 a 2a 3a 2 bc d Finda, b,c, d if X, Y are independent, and P(X = 2) = 2P(X = 1)
Random variables X and Y have joint distribution given by the following table:

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