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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.Repeat Example 1a for the function f(x)=2x2 on [1,2]. What is the probability that X will be between 32 and 2?3. Let X Pois(4). Find the distribution of Y = √X.
- Suppose a sample Y1, ..., Yn is selected from an exponential distribution with E (Yi) = θ and V (Yi) = θ2. Let θˆ be an estimator of the mean, θ.1. Explain what a parameter is. What is the parameter of interest in this case?2. Name the properties of a good estimator and explain, shortly, what is meant by each. (a)(b) (c)3. Given the properties you discussed in (2) and the parameter of interest identified in (1), what estimator would you propose?4. What would the sampling distribution of this estimator be? Why?1Example 23. A distribution x1, X2,..., Xp., Xp with frequencies, fp f2,..., f..., fn is transformed into the distribution y1, y2,.….., Y,.., yn with the same corresponding frequencies by the relation y, = ax, + b, where a and b are constants. Show that the mean, median and mode of the new distribution are given in terms of the first distribution by the same transformation.9. Assume that X has an exponential distribution. If Var (4-4)=20. Then E(X) is a. 20 b.0 c. 5 d. 10 e. None 10. The variable X has a uniform distribution and
- 5.2.1 The conditional probability distribution of Y given X = x is fy.0) = xe-y for y> 0, and the marginal probability distribution of X is a continuous uniform distribution over 0 to 10. a. Graph fyv) = xe¬w for y > 0 for several values of x. Determine: b. P(Y < 2|X = 2) d. E(Y |X = x) f. fr(y) c. E(Y | X = 2) e. fxy(x, y)1.2. Assume that Billy's driving time to class is uniformly distributed from 15-20 minutes in good weather, and uniformly distributed from 25-30 minutes in bad weather. Let the RV X describe Billy's drive time. If the probability of good weather is 2/3. a) .describe mathematically the PDF of X. b) ..describe graphically the PDF of X. c) .find the probability that Billy's drive time is >18 minutes. d) .find the Expected Value of Billy's drive time, irrespective of the weather.EX7.8) Let Y be a random variable having a uniform normal distribution such that Y U(2,5) 2 Find the variance of random variable Y.
- 6. Find the mean and variance of fx (x) = 2e eA* la.2...) (x) = de1 b) For any data set, approximately 95% of the observations fall in the interval (T − 2s, T+2s). True FalseQ.7 For an exponential distribution fx(x; 4) = {4e*; Sue Hx, x 2 0 e.w. where u > 0 Write (Don't derive) (a) Mean (expected value), i.e. EX). (b) Standard deviation, i.e. S. D. (X). (c) Moment generating function, i.e. Mx(t). (d) Cumulative distribution function (cdf), Fx(x). of Nizy