Let Y have probability density function fY(y)={ (2(θ-y))/(θ^2), 0<y<θ 0, elsewh

step 1 For exercise 44 part A, we first see that f sub y of x is equivalent to 2x divided by theta minu

Let Y have probability density function fY(y)={ ...

Let $Y$ have probability density function $$f_{Y}(y)=\left\{\begin{array}{ll} \frac{2(\theta-y)}{\theta^{2}}, & 0<y<\theta \\ 0, & \text { elsewhere } \end{array}\right.$$ a. Show that $Y$ has distribution function $$F_{Y}(y)=\left\{\begin{array}{ll} 0, & y \leq 0 \\ \frac{2 y}{\theta}-\frac{y^{2}}{\theta^{2}}, & 0<y<\theta \\ 1, & y \geq \theta \end{array}\right.$$ b. Show that $Y / \theta$ is a pivotal quantity. c. Use the pivotal quantity from part (b) to find a $90 \%$ lower confidence limit for $\theta$

Let $Y$ have probability density function $$f_{Y}(y)=\left\{\begin{array}{ll}
\frac{2(\theta-y)}{\theta^{2}}, & 0<y<\theta \\
0, & \text { elsewhere }
\end{array}\right.$$ a. Show that $Y$ has distribution function $$F_{Y}(y)=\left\{\begin{array}{ll}
0, & y \leq 0 \\
\frac{2 y}{\theta}-\frac{y^{2}}{\theta^{2}}, & 0<y<\theta \\
1, & y \geq \theta
\end{array}\right.$$ b. Show that $Y / \theta$ is a pivotal quantity.
c. Use the pivotal quantity from part (b) to find a $90 \%$ lower confidence limit for $\theta$

Key Concepts

Pivotal Quantity

A pivotal quantity is a function of the sample data and unknown parameters that has a probability distribution independent of the parameters. This property makes pivotal quantities extremely valuable for statistical inference, especially in constructing confidence intervals and hypothesis tests, since critical values and probabilities can be derived without needing to know the true parameter values.

Confidence Interval Construction

Confidence interval construction is a method in statistical inference used to estimate a range of values within which an unknown parameter is expected to lie with a certain probability. By using functions like pivotal quantities, one can create intervals that have a guaranteed coverage probability, such as a 90% lower confidence limit, meaning that there is a 90% chance the interval will contain the true parameter value.

Cumulative Distribution Function (CDF)

The CDF is a fundamental concept in probability theory that gives the probability that a random variable is less than or equal to a given value. It is obtained by integrating the probability density function (PDF) over the range from the lower bound up to the point of interest. The CDF is particularly useful for determining probabilities over intervals and for serving as a basis in transforming between probability scales and observed data.

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