Concept explainers
Consider a random sample X1,..., Xn from the
where −1 ≤ θ ≤ 1 (this distribution arises in particle physics). Show that
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Chapter 6 Solutions
Probability and Statistics for Engineering and the Sciences
- Respiratory Rate Researchers have found that the 95 th percentile the value at which 95% of the data are at or below for respiratory rates in breath per minute during the first 3 years of infancy are given by y=101.82411-0.0125995x+0.00013401x2 for awake infants and y=101.72858-0.0139928x+0.00017646x2 for sleeping infants, where x is the age in months. Source: Pediatrics. a. What is the domain for each function? b. For each respiratory rate, is the rate decreasing or increasing over the first 3 years of life? Hint: Is the graph of the quadratic in the exponent opening upward or downward? Where is the vertex? c. Verify your answer to part b using a graphing calculator. d. For a 1- year-old infant in the 95 th percentile, how much higher is the walking respiratory rate then the sleeping respiratory rate? e. f.arrow_forwardFind the average rates of change of f(x)=x2+2x (a) from x1=3 to x2=2 and (b) from x1=2 to x2=0.arrow_forwardSuppose X1, . . . , Xn are i.i.d. from a continuous distribution with p.d.f. fθ(x)=1/2(1+θx)if −1≤x≤1, where θ ∈ [−1, 1] is an unknown parameter. (a) Find E(X1).(b) Find the MME for θ. (c) Compute the variance of your MME from part (a).arrow_forward
- 2. Let X₁,..., Xn be iid observations from a pdf defined by 0 f(x|0) = 0 0. (1+x)¹+0¹ Find a complete sufficient statistic.arrow_forwardSuppose X1, . . . , Xn are i.i.d. from a continuous distribution with p.d.f. fθ(x) = 1/θ if 0 ≤ x ≤ θ, where θ > 0 is an unknown parameter. (a) Find E(X1). ANSWERED(b) Find the MME for θ. ANSWERED (c) Compute the variance of your MME from part (a). ANSWERED Can you estimate the standard error for your estimators?arrow_forwardSuppose X1, . . . , Xn are i.i.d. from a continuous distribution with p.d.f. fθ(x) = 1/θ if 0 ≤ x ≤ θ, where θ > 0 is an unknown parameter. (a) Find E(X1) ALREADY ANSWERED (b) Find the MME for θ. (c) Compute the variance of your MME from part (a). I have asked this question multiple times and this last time it asked for clarification of whether it was MME (Methods of Moments Estimator) or MLE (Maximum Likelyhood Estimator). It calls for MME not MLE.arrow_forward
- 9. Let Y1, Y2,..., Yn constitute a random sample from the pdf given by f (y) = 2 (0 – y) for 0 < y <0. %3D Find an estimator for 0 by using the method of moments. estimator =arrow_forward2 Consider an SRS 2Yi3;=, from the pdf fy(y;o)= -(y-0) for y≤ [0,00). ye e What is an unbiased estimator involving Ymin?arrow_forwardLet fo(X) = {(1/7) 0 X₁, X₂,..., Xn be i.i.d. samples with pdf 0 0, Otherwise. Assume: 1 Estimator 1 is Ô = 2 X. 2 Estimator 2 is Ô = X(n). Answer the following questions: Q1 Are these estimators unbiased? Q2 Are these estimators consistent?arrow_forward
- 2. Let dt Find the average %3D value of f on the interval [o, 1arrow_forwardIf 1. If for >0 the energy of a pdf is E(x)=r, find the variance. (A) 1 (B) 2 (C) 1/2 (D) Undefined (E) 0arrow_forward• If the joint PDF of X and Y is shown as follows: fxx(x, y) = e-(x+y) Find the following: 12) The marginal distributions of X and Y. 13) Find the value of P[X | 0 < Y < 5].arrow_forward
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