Science:Math Exam Resources/Courses/MATH 180/December 2017/Question 11 (e)

MATH 180 December 2017

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Question 11 (e)
The function $$P(x)={\frac {1}{4}}xe^{-x/2},\quad x\geq 0,$$ is an example of a Gamma distribution, a function that may be used to describe certain probabilities. (e) Draw a large graph of ${\textstyle P(x)}$ below, incorporating all of the features determined in the previous parts of this question.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Noting that the domain of $P$ is given by $\{x\geq 0\}$, draw its graph based on the information in part (a)-(d). You can split the domain into three parts, $[0,2)$, $[2,4)$, and $[4,\infty )$, and draw each piece of the graph.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Note that the domain of $P$ is given as $\{x\geq 0\}$. Part (a) tells us that the graph of $P$ passes through $(0,0)$. From part (b), we get the horizontal asymptote of $P$ as $y=0$. In other words, as $x$ goes to infinity, the graph approaches to $y=0$, i.e., $x$-axis. From part (c) and (d), we see that $P$ is increasing on $[0,2)$ and decreasing on $(2,\infty )$, while it is concave down on $[0,4)$ and concave up on $(4,\infty )$. Collecting the information, we can draw the graph of $P$ as follows.