Let X be an exponential random variable with standard deviation σ. Find P(|X − E(X)| > kσ ) for k = 2, 3, 4, and compare the results to the bounds from Chebyshev’s inequality.
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Let X be an exponential random variable with standard deviation σ. Find
P(|X − E(X)| > kσ ) for k = 2, 3, 4, and compare the results to the bounds
from Chebyshev’s inequality.
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- Suppose that Y is an exponential random variable with λ = 4. Find. P[Y > (E(Y) + 2√√Var(Y))] . (round to 4 decimal points)Let X1, X2, ..., Xn be a random sample from a normal distribution with mean u and variance o?. Find an unbiased estimator for o? and show that ΣΧ-Χ- E(X; - X)² = E(X²) – nX². i=1 i=1A random sample of size n1 = 15 is selected from a normal population with a mean of 75 and a standard deviation of 9. A second random sample of size n2 = 9 is taken from another normal population with mean 69 and standard deviation 15. Let X1 and X2 be the %3D two sample means. Find: (a) The probability that X - X2 exceeds 3. (b) The probability that 4.9 < X – X2 < 5.9. Round your answers to two decimal places (e.g. 98.76). (a) i (b)
- f (x) = 4xc for 0 ≤ x ≤ 1 ; Find the constant c so that f (x) is apdf of some random variable X, and then find the cdf, F (x) = P(X ≤ x). Sketch graphsof the pdf f (x) and the cdf F (x), and find the mean μ and variance σ2.Q1. let Y₁ < Y₂ < Y3 < Y4 < Y5 are the order statistics of the random sample of size 5 from the distribution : f(x) = 3x², 0Let a random variable X has the following function:FX(x) = { 0 ; x < 0(1/2)+(x/2) ; 0 ≤ x < 11 ; x ≥ 1. Verify that FX(x) is a CDF. Determine (i) the PDF of X, (ii) E[e^X], (iii) P[X = 0|X≤ 0.5].2. Suppose that X1, . Xn are iid Geometric random variables with frequency func- ... tion f(x; 0) = 0(1 – 0)", x = 0, 1, 2, ..., 0 E (0, 1). Find the ML estimator 0, of 0. Show that Ôn is consistent an find its asymptotic distribution.A random variable, X, has a pdf given by The expected value of Y is O a. 3/5 O b. 5/3 O c. 7/5 O A random variable Y, is related to X by Px(x) d. 7/3 = 1 -3 < x < 1. and 0 otherwise 4 Y = X²A random sample of size n₁ = 14 is selected from a normal population with a mean of 76 and a standard deviation of 7. A second random sample of size n₂ = 9 is taken from another normal population with mean 71 and standard deviation 11. Let X₁ and X₂ be the two sample means. Find: (a) The probability that X₁ – X₂ exceeds 4. 1 2 (b) The probability that 4.3 ≤ X₁ – X2 ≤ 5.6. Round your answers to two decimal places (e.g. 98.76). (a) i (b) i3. Suppose X is a discrete random variable with pmf defined as p(x) = log10 ( for %3D x = {1,2,3,...9} Prove that p(x) is a legitimate pmf.Answer no. 1 onlyLet X be a continuous random variable with p.d.f. f(x) and distribution function F(x). If Y = X², (a) what is the g(y)? 14/12 (b) If ƒ(x) = -√2/² 2π -∞Recommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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