Answer to Question #185092 in Statistics and Probability for SARALA DEVI

Question

1-Let n X₁, X₂ ....X_n be random sample of size n from a distribution with probability

density function f(x,q)=qx^q-1

0, else where.

Obtain a maximum likelyhood

estimator of q.

2- Let , X₁, X₂ Xn be independently and identically distributed b(1, p) random

variables. Obtain a confidence internal for p using Chebychev’s inequality

Answer 1

1. The given Probability density function is-

$f(x,q)=qx^{q-1}$

Taking log on both sides-

$log f(x,q)=logqx^{q-1}\\logf(x,q)=logq+(q-1)logx$

Log liklihood function is given by-

$L(X_1,X_2,X_3,...,X_n)=logf(x,q)=logq+(q-1)log x$

The maximum liklihood estimator of q-

$\phi_{ML}=\phi_{ML}(X_1,X_2,.....,X_n)$

2.Confidence interval by chebychev,s inequality is-

$P(|\dfrac{1-p}{n}|)<\dfrac{\sigma}{E^2}$

$P(|(X-E[X])|\ge k\sigma)\le \dfrac{1}{k^2}$