Let X₁, X₂,..., X, be mutually independent and identically distributed random variables with means and variance o². Let X=(-1X)/n. Show that Σ(X₂ X)²-(X₂ µ)² n(X μ)² km1 km1 216 [2(X-XX]- and hence E (n-1)0². EXPECTATION
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![4. Let X₁, X₂,..., X be mutually independent and identically distributed random
variables with means μ and variance o². Let X = (-1X)/n. Show that
71
12
Σ(X₂ X)² = (X₂-µ)²= n(X-µ)²
k=1
k=1
216
n
EX(X-X)² = (n − 1)0².
(-X)²] =
_k=1
and hence
EXPECTATION](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F635941ec-c946-4839-83d0-71c4820f36e4%2F1a78bc61-3e58-4a5c-b1a2-a910abf80356%2Fcd180ij_processed.jpeg&w=3840&q=75)
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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.Suppose that f(x) = 0.125x for 0 < x < 4 Determine the mean and the variance of X.Show that variance o? = (x²) – ((x))²
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- Let x1, x2, ..., n represent a random sample from a distribution with mean E(x) and variance Var(x). Show that Cov(x, x₁ - x) = 0.Let X be a continuous random variable with PDF 3 x > 1 x4 fx(x) = otherwise Find the mean and variance of x.Let X1 be a normal random variable with mean 2 and variance 3 and let X2 be a normal random variable with mean 1 and variance 4. Assume that X1 and X2 are independent. (a) What is the distribution of the linear combination Y = 2X1 +3X,? (b) What is the distribution of the linear combination Y = X, – X,?
- If X is a random variable with pdf f(x) = 2x − 2 where x = (1, 2), find the variance of Y = 2X - 3.Let X₁, X2, X3, Xn be a random sample with unknown mean EX; = µ, and unknown variance Var(X₂) = o². Suppose that we would like to estimate 0 = μ². We define the estimator as 2 • - (™)² - [ 2x]* Xk to estimate 0. Is an unbiased estimator of ? Why?Let X be a random variable and a real number. Show that E(X - a)² = varX + (µ − a)² Hereμ = EX is the expected value of the random variable X and varX = E(X - μ)^2 is the variance of the random variable X. Guidance: start from the representation - (X-a)^2 = (X µ + μ- a)^2 and group the right side of the representation appropriately into the form (Z + b)^2, where Z is some random variable and b is a real number and open the square. The task should be solved with the help of the expected value calculation rules.