QUESTION 3 Given the probability distribution for 1 < X< oo (a) What is the MLE...
3. Let X1, X2, . . . , Xn be a random sample from a distribution with the probability density function f(x; θ) (1/02)Te-x/θ. O < _T < OO, 0 < θ < 00 . Find the MLE θ
-wa exp{-(20 )2}, where The Normal(μ,02) distribution has density f(x) -oo < μ < oo and σ > 0. Let the randon variable T be such that X-log(T) is Normal(μ, σ2). Find the density of T. This distribution is known as the log normal Do not forget to indicate where the density of T is non-zero. 10.
4. For the distribution f(z; a) = (a +1)xº, 0<x<1, what is the MLE of a based on a random sample X1, X2, ..., Xn?
6.2.1 2. Recall that θ--r/ Σ (θ, 1 ) distribution. Also, W - i-1 log Xi has the gamma distribution Г(n, 1/ ) -1 log X, is the mle of θ for a beta (a) Show that 2θW has a X2(2n) distribution. (b) Using part (a), find ci and c2 so that (6.2.35) for 0 < α < 1 . Next, obtain a (1-a) 100% confidence interval for θ.
X, be a random sample from a distribution with the probability density function f(x; θ) = (1/02).re-z/. 0 <エく00, 0 < θ < oo. Find the MLE θ
Let f(x; θ) = 1 θ x 1−θ θ for 0 < x < 1, 0 < θ < ∞. (1) Show that ˆθ = − 1 n Pn i=1 log(Xi) is the MLE of θ. (2) Show that this MLE is unbiased. Exactly 6.4-8. Let f(x:0)-缸붕 for 0 < x < 1,0 < θ < oo 1 1-0 (1) Show that θ Σ-1 log(X) is the MLE of θ (2) Show that this MLE is unbiased.
3. This problem is concerned with the maximum likelihood estimate (MLE) of various distributions. Bob, Céline and Daisy want to model the distribution of the heights of 20 students in the classroom. They get the following data (in cm) : 168, 177, 194, 169, 159, 172, 174, 177, 159, 172, 181, 171, 168, 162, 168, 157, 180, 174, 162, 177. (i) Bob took Math170A, and he wants to model the heights by the normal distribution with probability density p(x) e...
2-2.3 A probability distribution function for a random variable has the form F,(x) = A(1-exp[-(x-1)) 1 < x < oo -00<xs1 a) For what value of A is this a valid probability distribution function? b) What is Fx (2)? c) What is the probability that the random variable lies in the interval 2 X < 00? d) What is the probability that the random variable lies in the interval 1 <X s3?
Let Xi, X2, Xn be ar ensity function f(r; θ) = (1/2)e-Iz-이,-oo < x < 00,-00 < θ < oo. Find the d MLE θ
2. Let Xi, X2, . Xn be a random sample from a distribution with the probability density function f(x; θ-829-1, 0 < x < 1,0 < θ < oo. Find the MLE θ