Find the density of cX when X follows a gamma distribution. Show that only $\lambda$ is affected by such a transformation, which justifies calling $\lambda$ a scale parameter.

A supplier of kerosene has a weekly demand Y possessing a probability density function given by

$$f ( y ) = \left\{ \begin{array} { l l } { y , } & { 0 \leq y \leq 1 } \\ { 1 , } & { 1 < y \leq 1.5 } \\ { 0 , } & { \text { elsewhere, } } \end{array} \right.$$

with measurements in hundreds of gallons. The supplier’s profit is given by

$$U = 10 Y - 4$$

. Find the probability density function for U.

Suppose that a random variable Y has a probability density function given by

$$f ( y ) = \left\{ \begin{array} { l l } { 6 y ( 1 - y ) , } & { 0 \leq y \leq 1 } \\ { 0 , } & { \text { elsewhere. } } \end{array} \right.$$

Find F(y).

Suppose that a random variable Y has a probability density function given by

$$f ( y ) = \left\{ \begin{array} { l l } { k y ^ { 3 } e ^ { - y / 2 } , } & { y > 0 } \\ { 0 , } & { \text { elsewhere. } } \end{array} \right.$$

What is the probability that Y lies within 2 standard deviations of its mean?

Suppose that a random variable Y has a probability density function given by

$$f ( y ) = \left\{ \begin{array} { l l } { k y ^ { 3 } e ^ { - y / 2 } , } & { y > 0 } \\ { 0 , } & { \text { elsewhere. } } \end{array} \right.$$

. What are the mean and standard deviation of Y?

Suppose that a random variable Y has a probability density function given by

$$f ( y ) = \left\{ \begin{array} { l l } { 6 y ( 1 - y ) , } & { 0 \leq y \leq 1 } \\ { 0 , } & { \text { elsewhere. } } \end{array} \right.$$

Graph F(y) and f (y).

The Weibull cumulative distribution function is $F(x)=1-e^{-(x / \alpha)^{\beta}}, \quad x \geq 0, \quad \alpha>0, \quad \beta>0$ a. Find the density function. b. Show that if W follows a Weibull distribution, then $X=(W / \alpha)^{\beta}$ follows an exponential distribution. c. How could Weibull random variables be generated from a uniform random number generator?

With your group, review "The Return" to identify elements of foreshadowing and situational irony in the story. Then, individually, complete the charts to understand how foreshadowing sets up expectations that affect situational irony. Finally, gauge the impact of situational irony on what the story says about homecomings.

$$\begin{array}{|l|l|l|}\hline & \text{FORESHADOWING IN "THE RETURN"} & \\\hline \text{STORY CLUES} & \text{QUESTIONS RAISED} & \text{WHAT THE CLUES SUGGEST}\\ \hline \\ \hline \\ \hline \\ \hline & \text{SITUATIONAL IRONY IN "THE RETURN"} & \\ \hline \text{EXPECTATIONS} & \text{WHAT ACTUALLY HAPPENS} & \text{CONNECTION TO THE STORY'S MEANING}\\ \hline \\ \hline \\ \hline \\ \hline \end{array}$$

Find the indefinite integral. $\int 6 d x$

Suppose that a random variable Y has a probability density function given by

$$f ( y ) = \left\{ \begin{array} { l l } { k y ^ { 3 } e ^ { - y / 2 } , } & { y > 0 } \\ { 0 , } & { \text { elsewhere. } } \end{array} \right.$$

. Find the value of k that makes f (y) a density function.

A gas station operates two pumps, each of which can pump up to 10,000 gallons of gas in a month. The total amount of gas pumped at the station in a month is a random variable Y (measured in 10,000 gallons) with a probability density function given by

$$f ( y ) = \left\{ \begin{array} { l l } { y , } & { 0 < y < 1 } \\ { 2 - y , } & { 1 \leq y < 2 } \\ { 0 , } & { \text { elsewhere. } } \end{array} \right.$$

. Find the probability that the station will pump between 8000 and 12,000 gallons in a particular month.

A gas station operates two pumps, each of which can pump up to 10,000 gallons of gas in a month. The total amount of gas pumped at the station in a month is a random variable Y (measured in 10,000 gallons) with a probability density function given by

$$f ( y ) = \left\{ \begin{array} { l l } { y , } & { 0 < y < 1 } \\ { 2 - y , } & { 1 \leq y < 2 } \\ { 0 , } & { \text { elsewhere. } } \end{array} \right.$$

. Find F(y) and graph it.

A gas station operates two pumps, each of which can pump up to 10,000 gallons of gas in a month. The total amount of gas pumped at the station in a month is a random variable Y (measured in 10,000 gallons) with a probability density function given by

$$f ( y ) = \left\{ \begin{array} { l l } { y , } & { 0 < y < 1 } \\ { 2 - y , } & { 1 \leq y < 2 } \\ { 0 , } & { \text { elsewhere. } } \end{array} \right.$$

Graph f (y).

The U.S. population is approximated by the function $P(t)=\frac{616.5}{1+4.02 e^{-0.5 t}}$ where P(t) is measured in millions of people and t is measured in 30-year intervals, with t = 0 corresponding to 1930. What is the expected population of the United States in 2020 (t=3)?