Mathematical Statistics and Data Analysis
3rd Edition
ISBN: 9781111793715
Author: John A. Rice
Publisher: Cengage Learning
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Consider two assets (X and Y) with mX = 10%, mY = 10%, σX2=.16, σY2=.25, and Cov(X,Y) = -.125. What is the expected return and variance of the portfolio having 70% invested in X and 30% invested in Y? Compare the risk and return of this portfolio with the risks and returns associated with investing everything in either X or Y.
a) What is p(XY)?
b)
What is the expected return of the portfolio (m.7X+.3Y)?
c)
What is the standard deviation of the portfolio (s.7X+.3Y)?
d)
How does the standard deviation of the portfolio (s.7X+.3Y) compare to the standard deviations of assets X and Y?
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Suppose that X is an exponential random variable with mean 5. (The cumulative distribution function is F(x) = 1- e-x/5 for x >= 0, and F(x) = 0 for x < 0.
(a) Compute P(X > 5).
(b) Compute P(1.4 <= X <= 4.2).
(c) Compute P(1.4 < X < 4.2).
Let X be a Poisson random variable with E(X) = 3. Find P(2 < x < 4).
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- If Z is a discrete random variable with possible values of -1, 0, and 1, and the probability mass function is given by P(Z = -1) = 0.4, P(Z = 0) = 0.3, and P(Z = 1) = 0.3, what is the standard deviation of Z?arrow_forwardConsider a real random variable X with zero mean and variance σ2X . Suppose that wecannot directly observe X, but instead we can observe Yt := X + Wt, t ∈ [0, T ], where T > 0 and{Wt : t ∈ R} is a WSS process with zero mean and correlation function RW , uncorrelated with X.Further suppose that we use the following linear estimator to estimate X based on {Yt : t ∈ [0, T ]}:ˆXT =Z T0h(T − θ)Yθ dθ,i.e., we pass the process {Yt} through a causal LTI filter with impulse response h and sample theoutput at time T . We wish to design h to minimize the mean-squared error of the estimate.a. Use the orthogonality principle to write down a necessary and sufficient condition for theoptimal h. (The condition involves h, T , X, {Yt : t ∈ [0, T ]}, ˆXT , etc.)b. Use part a to derive a condition involving the optimal h that has the following form: for allτ ∈ [0, T ],a =Z T0h(θ)(b + c(τ − θ)) dθ,where a and b are constants and c is some function. (You must find a, b, and c in terms ofthe information…arrow_forwardGiven that the discrete random variables X has probability mass function, Pr(X=x)= (x/6 where x=1,2,3) 0 elsewhere, Describe and graph its cumulative distribution function,F(x)arrow_forward
- The diameter of a round rock in a bucket, can be messured in mm and considered a random variable X in f(x) f(x) = k(x-x4) if 0 ≤ x ≤ 1f(x) = 0 otherwise. Find the expectation E(4X+ 3) and variance V(4X+3)arrow_forwardX and Y are random variables with the following joint PMF. X?, Y? 6 -7 9 9 0.2 0.1 0.16 -3 0.1 0.15 ??? Calculate the marginal PMF for X. What is P(X = 9) * P(X = -3)? Hint: infer the bottom right corner first.arrow_forwardGiven that the discrete random variable X has probability mass function, Pr (X = x) = x/6 where x = 1, 2, 3 0 elsewhere, Describe and graph its cumulative distribution function, F(x).arrow_forward
- The probability function of the random variable X is defined as f(x)=cx²(1-x)² for 0<x<1, otherwise f(x)= 0. Calculate the constant c , the expected value and the variance.arrow_forwardSuppose that the random variables Y1 and Y2 have joint probability distribution function. f(y1, y2) = 2, 0 ≤ y1 ≤ 1, 0 ≤ y2 ≤ 1, 0 ≤ y1 + y2 ≤ 1, 0, elsewhere (a) Use R to calculate P(Y1 ≥ 1⁄6 | Y2 ≤ 1⁄5). (Round your answer to four decimal places.) P(Y1 ≥ 1⁄6 | Y2 ≤ 1⁄5) = (b) Use R to calculate P(Y1 ≥ 1⁄6 | Y2 = 1⁄5). (Round your answer to four decimal places.) P(Y1 ≥ 1⁄6 | Y2 = 1⁄5) =arrow_forwardWe require a non-zero variance in the central limit theorem, because Select one: a. otherwise, the Xi do not converge in distribution. b. otherwise we cannot use the weak law of large numbers and the central limit theorem gives a stronger conclusion. c. otherwise the Xi are constant and therefore do not converge to the normal distribution. d. otherwise the Xi are all equal to the constant random variable 0.arrow_forward
- (b) Let Z be a discrete random variable with E(Z) = 0. Does it necessarily follow that E(Z³) = 0? If yes, give a proof; if no, give a counterexample.arrow_forwardConsider a function F (x ) = 0, if x < 0 F (x ) = 1 − e^(−x) , if x ≥ 0 Is the corresponding random variable continuous?arrow_forwardlet x denote the number of failures before the third success in a sequence of independent and identical Bernoulli trials. Given a simple random sample of (4,5,6) to estimate the probability of success p, which statement is true? a. the method of moment estimate of p is 0.167 b. p is 0.2 c. cannot be estimated d. p is 0.375arrow_forward
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