.X, be an id sample from the distribution r >1 (a) Using this distribution, find Eflog(X)...
4. A sample of size n-81 is taken from an exponential distribution with the pdf f(x)-Be-6x, θ > 0, x > 0. The sample mean is i-35. Find a 95% large- sample confidence interval for θ using the Central Limit Theorem.
Find the MME for r and λ for the Gamma distribution given by
fX(x; r, λ) = λ r Γ(r) x r−1 e −λx where x > 0, r > 0, and λ
> 0. Assume a random sample of size n has been drawn
ar-le-k 4. Find the MME for r and λ for the Gamma distribution given by fx(z;r, A) where x > 0, r > 0, and λ 〉 0, Assume a random sample of size n...
9. Let the distribution of X for r>0 be Random Variables and Distribution Functions 70 What is the density function of X for r >0?
2.5.9. The random variable X has a cumulative distribution function for xo , for xsO . for r>0 F(x) = z? 1 +x2 Find the probability density function of X.
(d). Let X, X,...,x be a random sample from the Normal(0,0) distribution, 0 >0. Find the uniformly most powerful test for H:050 versus H,:0>
2. Let X 1, , Xn be iid from the distribution modeled by 8-2 fx (1:0)-(9. θ):r"-"(1-2) dr where 0 < x < 1 and θ > 1 Find the MME (method of moments estimate/estimator) for 0
10. Consider this joint pdf. c(r+ y 0 otherwise (a) Find c. (b) Find frv). (c) Find fyy) (d) What is the probability that x > 0 giveny-1?
Written Problems l. Let Yı, Ya, Ya be a random sample from an Exponential distribution with density function f(y)-Te-3, y > 0. Find the MSE of each of the following estimators of θ: (a)-互华 (c) θ=F 2
The velocity of a particle in a gas is a random variable X with probability distribution fx(x) = 27 x2 -3x x >0. The kinetic energy of the particle is Y = {mXSuppose that the mass of the particle is 64 yg. Find the probability distribution of Y. (Do not convert any units.)
The probability mass function of a random variable X is given by Px(n)r n- (a) Find c (Hint: use the relationship that Ση_0 n-e) (b) Now assume λ = 2, find P(X = 0) (c) Find P(X>3)