Y₁ (a) Find the joint pdf of (V,, V₁, V₂), where V₁ V₂ and v₁ - (b) Find the marginal pdf of V₂.
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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.Let X and Y have the joint pdf f(x,y)= x+y , 0<=x<=1, 0<=y<=1. Calculate the mean(x) mean (y) variance (x) variance(y)Consider (Y, X) values in a population as (8, 2), (9, 3), (10, 3), (12, 5), (14, 6), (16, 6) and (17, 8). Draw all possible samples from this population and compute Ratio estimator for each sample. Show that the ratio estimator is a biased estimator of population mean but has smaller mean square error as compared with the variance of mean per unit estimator.
- A random variable Y has a uniform distribution over the interval (8,, e,). Derive the variance of Y. Find E(Y)? in terms of (e,, e,). E(Y)? = Find E(Y) in terms of (e,, e,). E(Y) = Find V(Y) in terms of (8,, e,). V(Y) =Suppose X and Y are two random variables with covariance Cov(X, Y) = 3 and Var(X) = 16. Find the correlation coefficient between X and Y.If(x,y)-N(0,-11,4,0.5),the marginal distribution of X- Y-
- Let Y1, . . . , YN be a random sample from the Normal distribution Yi ∼ N(ln β, s2) where s2 is known. Find the Score function, the estimating equation and the information matrix.Let a and b be any real numbers and let x = (x1,Xz -- -- Xp m-, X) be a data set with a sample mean f and a sample variance s. Show that if y, = (ax, + b), then S3 = a*s;Let Y, represent the th normal population with unknown mean 4, and unknown variance of for i=1,2. Consider independent random samples, Y₁, Y2. Yin, of size n₁, from the ith population with sample mean Y, and sample variance S² = ₁₁-1(Y₁-₁². j=1 (g) For non-zero constants a's, what is the distribution of U₂ = a₁Y₁-a₂Y₂? State all the relevant parameters of the distribution. (h) Find the standard error of U₂ in part (g), assuming that of = 0² = 0². (i) Discuss how the distribution of Y₁ - ₂ can be used to test the equality of the two population means, #₁ and μ2, when o² = 0 = 0² is known. (j) Define appropriate rejection regions, in terms of Y₁ - Y2, for testing Ho: #₁ = 2 against a two-sided alternative hypothesis at the a level of significance.
