By using a change of variables, verify that the univariate Gaussian distribution given by 1 1 N(x|μ, σ²) = (200²)1/2 exp{- 202 (x − μ)²} 20² satisfy ∞ E[x] = N(x|μ₁0²)xdx = μ -∞
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![By using a change of variables, verify that the univariate Gaussian distribution given by
1
1
N(x|μ, σ²) =
exp{-
{ - 2 / 2 ( x − μ)²}
(2πσ²)¹/2 exp
satisfy
SON
∞
E[x] =
N(x|μ, σ²)xdx = μ](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5c96556e-466a-4720-8c5a-697057e0ec90%2F43b08006-2c13-4aa8-92bb-67316993d081%2Ffr854kfn_processed.png&w=3840&q=75)
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- Consider the Weibull distribution with pdf fx (x) = 3x²e¬**, x > 0. Derive the Hazard rate function (failure rate) fx (t) R (t) : %3D 1- Fx (t)a) You throw two fair dice labeled 1, . . . , 6 independently of each other and let X be the minimum of the two results. Calculate E[X2]. (b) Suppose that X has pdf given by f(x) = 3x^(−4) if x > 1 and by f(x) = 0 if x <= 1. Calculate the median of X.The moment generating function of the exponential distribution with rate A is m(t) = for t < A. Moreover, skewness of a random variable X is defined as X – E(X) E /var(X) Find the skewmess for the standard exponential distribution.
- Suppose Xn is the sample mean of a random sample of size n from a distribution that has a Gamma(2,1/2) (where a = 2, B = }) pdf f(x) = 4xe-2a, 0 < x < ∞, zero elsewhere. Use the Central Limit Theorem to deduce that the random variables Vn(Xn – 1) converges in distribution to N(0, 1/2).Q5 Find the variance for the PDF px(x) = e-«/2, x > 0.A continuous random variable X has a P.D.F. fx (x) = 3 X, 0Sr a) and (ii) P (x > b) = 0.05.
- Assume the time required to fully recharge an electric car is uniformly distributed between 160 and 220 minutes. What is the expected charge time, E(x)?Which of the following is the correct sum of squared residuals to be minimized to find tne fit f(x) = £ (2r-1)"* - using a data set ((xo, Yo), (x1,y1), --, CKn. Yn)} ? A.) Eo(In(y;) + a – b2x; + 1)² B.) E-(In(y;) + In a + b In(2x; – 1))² D.) E7-o(In(y;) – In a – b In(2x; – 1))² E.) E-o(In(y;) + In a – b2x; + 1)²(b) Let X represent the lifetime in months of a certain brand of light bulb. The pdf of X is given by f(x)=K(6-x), 0A variable X has a probability function f(x, 0) = 0e 0x; x≥0 y >0. The parameter 0 is estimated using two different estimators of simple random samples of size two: 01 2x+4x2 and 2=4x+5x2 3 3 Select the best estimator for \theta by comparing the Mean Squared ErrorsEL 466 416 13.) The continuous random variable (RV) X is uniform over [0,1). Given Y = -ln X what is P({0PLease Show all work so I can UnderstandSEE MORE QUESTIONSRecommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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