Question

7. Let $X_1, X_2, ..., X_n$ be a random sample of size $n$ from a distribution with the following probability density function. $f(x; \theta) = (\theta + 1)(1 - x)^{\theta}$, for $0 < x < 1$, $\theta > -1$ (a) Find the maximum likelihood estimator of $\theta$, $\hat{\theta}$. 8. Let $X_1, X_2, ..., X_n$ be a random sample of size $n$ from a distribution with the following probability density function. $f(x; \theta) = (\theta + 1)(1 - x)^{\theta}$, for $0 < x < 1$, $\theta > -1$ (a) Find the method of moments estimator of $\theta$, $\hat{\theta}$. (b) Calculate the maximum likelihood estimate of $\theta$, $\hat{\theta}$, when $x_1 = 0.2$, $x_2 = 0.4$, $x_3 = 0.5$, $x_4 = 0.9$. (b) Calculate the method of moments estimate of $\theta$, $\hat{\theta}$, when $x_1 = 0.1$, $x_2 = 0.1$, $x_3 = 0.2$, $x_4 = 0.3$.

          7. Let $X_1, X_2, ..., X_n$ be a random sample of size $n$ from a distribution with the following
probability density function.
$f(x; \theta) = (\theta + 1)(1 - x)^{\theta}$, for $0 < x < 1$, $\theta > -1$
(a) Find the maximum likelihood estimator of $\theta$, $\hat{\theta}$.
8. Let $X_1, X_2, ..., X_n$ be a random sample of size $n$ from a distribution with the following
probability density function.
$f(x; \theta) = (\theta + 1)(1 - x)^{\theta}$, for $0 < x < 1$, $\theta > -1$
(a) Find the method of moments estimator of $\theta$, $\hat{\theta}$.
(b) Calculate the maximum likelihood estimate of $\theta$, $\hat{\theta}$, when
$x_1 = 0.2$, $x_2 = 0.4$, $x_3 = 0.5$, $x_4 = 0.9$.
(b) Calculate the method of moments estimate of $\theta$, $\hat{\theta}$, when
$x_1 = 0.1$, $x_2 = 0.1$, $x_3 = 0.2$, $x_4 = 0.3$.

7. Let X1, X2, ..., Xn be a random sample of size n from a distribution with the following
probability density function.
f(x; θ) = (θ + 1)(1 - x)^θ, for 0 < x < 1, θ > -1
(a) Find the maximum likelihood estimator of θ, θ̂.
8. Let X1, X2, ..., Xn be a random sample of size n from a distribution with the following
probability density function.
f(x; θ) = (θ + 1)(1 - x)^θ, for 0 < x < 1, θ > -1
(a) Find the method of moments estimator of θ, θ̂.
(b) Calculate the maximum likelihood estimate of θ, θ̂, when
x1 = 0.2, x2 = 0.4, x3 = 0.5, x4 = 0.9.
(b) Calculate the method of moments estimate of θ, θ̂, when
x1 = 0.1, x2 = 0.1, x3 = 0.2, x4 = 0.3.

Added by Kristin S.

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