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lecture. Suppose that we have uniformly distributed data from Unif(0,$\theta$) and want to estimate $\theta$; we already know that the MLE for this is to take the maximum value of our samples. Suppose that we have a random sample $X_1, X_2, ..., X_n$ from this distribution and let $M$ be the maximum of these $n$ random variables. (a) Suppose we just take one sample from Unif(0,$\theta$) and get 327. Explain why (335.38, 13080) is a 95% confidence interval for $\theta$ by computing and $P(X_1 \le 327 | \theta = 335.38)$ $P(X_1 \le 327 | \theta = 13080)$. (b) Show that when $0 \le t \le 1$, $P(\frac{M}{\theta} \le t) = t^n$. (c) Solve the equation $P'(\frac{M}{\theta} \ge a) = 0.05$ for a. (d) Suppose we make 4 observations and find that the maximum observed value is 4.2. Use your result from the previous part to find a 95% one-sided confidence interval for $\theta$ of the form $(x,\infty)$. (e) What if we had n = 100 observations and an observed maximum of 4.2 instead? How would the confidence interval change?

          lecture. Suppose that we have uniformly distributed data from Unif(0,$\theta$) and want
to estimate $\theta$; we already know that the MLE for this is to take the maximum value
of our samples. Suppose that we have a random sample $X_1, X_2, ..., X_n$ from this
distribution and let $M$ be the maximum of these $n$ random variables.
(a) Suppose we just take one sample from Unif(0,$\theta$) and get 327. Explain why
(335.38, 13080) is a 95% confidence interval for $\theta$ by computing
and
$P(X_1 \le 327 | \theta = 335.38)$
$P(X_1 \le 327 | \theta = 13080)$.
(b) Show that when $0 \le t \le 1$,
$P(\frac{M}{\theta} \le t) = t^n$.
(c) Solve the equation
$P'(\frac{M}{\theta} \ge a) = 0.05$
for a.
(d) Suppose we make 4 observations and find that the maximum observed value is
4.2. Use your result from the previous part to find a 95% one-sided confidence
interval for $\theta$ of the form $(x,\infty)$.
(e) What if we had n = 100 observations and an observed maximum of 4.2 instead?
How would the confidence interval change?

$lecture. Suppose that we have uniformly distributed data from Unif(0,θ) and want to estimate θ; we already know that the MLE for this is to take the maximum value of our samples. Suppose that we have a random sample X1, X2, ..., Xn from this distribution and let M be the maximum of these n random variables. (a) Suppose we just take one sample from Unif(0,θ) and get 327. Explain why (335.38, 13080) is a 95% confidence interval for $\theta$ by computing and P(X1 ≤ 327 | θ = 335.38) P(X1 ≤ 327 | θ = 13080). (b) Show that when 0 ≤ t ≤ 1, P((M)/(θ)≤ t) = t^n. (c) Solve the equation P'((M)/(θ)≥ a) = 0.05 for a. (d) Suppose we make 4 observations and find that the maximum observed value is 4.2. Use your result from the previous part to find a 95% one-sided confidence interval for θ of the form (x,∞). (e) What if we had n = 100 observations and an observed maximum of 4.2 instead? How would the confidence interval change?$

Added by Briana L.

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