(i) If X has an exponential distribution with parameter 1, find the pdf of Y = VX . (ii) If X has a Cauchy's distribution with parameter a, prove that Y = 1/X has also a Cauchy's distribution with parameter 1/a.
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- 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.The following table provides values of the function f(x,y). However, because of potential; errors in measurement, the functional values may be slightly inaccurately. Using the statistical package included with a graphical calculator or spreadsheet and critical thinking skills, find the function f(x,y)=a+bx+cy that best estimate the table where a, b and c are integers. Hint: Do a linear regression on each column with the value of y fixed and then use these four regression equations to determine the coefficient c. x y 0 1 2 3 0 4.02 7.04 9.98 13.00 1 6.01 9.06 11.98 14.96 2 7.99 10.95 14.02 17.09 3 9.99 13.01 16.01 19.02Find the marginal PDFs of X and Y.
- = 1. (Review) (a) Suppose that X ~ Cauchya and c> 0. Show that Y cX Cauchyca (Hint: This is change of variable problem. First find the cdf for Y, then find the pdf by differentiating) (b) If X₁, ..., Xn are i.i.d. and X; ~ Cauchya, then Sn~ Cauchya (The point: Unlike in the various versions of the law of large numbers, Sn does not converge to a constant in any sense).How do I find an unknown positive normalizing constant, given x is a continuous unknown and the PDF of x?6. (Sec. 5.1) Two headlights of a car have the following joint pdf for their useful lifetimes X (the left headlight) and Y (the right headlight) ze(y+1) for r> 0.y > 0 f(x, y) 0 otherwise (a) What is the probability that the lifetime X of the left headlight exceeds 2.8? (b) Find the marginal pdfs of X and Y. Are the two lifetimes independent? Justify your answer (c) What is the probability that the lifetime of at least one headlight does not exceed 2.8?
- Let X and Y be independent and normally distributed RVs with the same PDF Find the PDFS of: (i) (ii) x?: (iii) x²+ Y?. X + Y;3. Joint distribution for continuous X, Y. Let X1, X2 denote two independent exponential r.v.'s with X; ~ Exp(1). Let X1 %3D X1 + X2 Find fy(y). Hint: recall the pdf for an exponential r.v., f(x) = de-x.Let {Xi} be i.i.d with the pdf f(x; θ) =c/x^(θ+1) x > 1, where θ > 0 is the unknown parameter. (a) Determine c in term of θ. (b) Given a sample {xi}i=1,...,n, find the MLE estimator θm= θm(x1, . . . , xn) for θ. (c)Let Z =X1^(-θ)+...+Xn^(-θ), Prove that θm and Z are independent.
