Question

5.1 - 6.2 (ch. 5 - continuous random variables and ch. 6 - the normal distribution)
read each question carefully and follow all instructions exactly. show all work; otherwise no credit. round to 2 decimals. draw a picture for each problem where it is relevant.

1. class passing time is the time between the classes. this is the time students can use in between their successive classes to leave from one class to go to another class. class passing time for our college is eight minutes. in other words, if x = # of minutes for class passing time, then 0 ≤ x ≤ 8.

(a) draw a uniform distribution.
(b) find the height (as a simplified fraction).
(c) find the area between 0 < x < 2.
(d) find the area for x = 3.

Explanation:

Step1: Recall uniform - distribution formula

For a uniform distribution $U(a,b)$ (here $a = 0$, $b = 8$), the probability density function $f(x)=\frac{1}{b - a}$ for $a\leq x\leq b$ and $0$ otherwise.

Step2: Draw the uniform distribution

The uniform distribution $U(0,8)$ has a rectangular - shaped graph. The $x$ - axis represents the class passing time $x$ with $0\leq x\leq8$, and the $y$ - axis represents the probability density $f(x)$. The base of the rectangle is from $x = 0$ to $x = 8$ on the $x$ - axis.

Step3: Calculate the height

Using the formula $f(x)=\frac{1}{b - a}$, substituting $a = 0$ and $b = 8$, we get $f(x)=\frac{1}{8-0}=\frac{1}{8}$.

Step4: Calculate the area between $0\lt x\lt2$

The area under the probability density function of a uniform distribution over an interval $[c,d]\subseteq[a,b]$ is given by $A=\frac{d - c}{b - a}$. Here, $c = 0$, $d = 2$, $a = 0$, $b = 8$, so $A=\frac{2-0}{8-0}=\frac{2}{8}=\frac{1}{4}=0.25$.

Step5: Calculate the area for $x = 3$

For a continuous random variable, the probability at a single point is $0$. So the area for $x = 3$ is $0$.

Answer:

(a) Draw a rectangle with base from $x = 0$ to $x = 8$ on the $x$ - axis and height $\frac{1}{8}$ on the $y$ - axis.
(b) $\frac{1}{8}$
(c) $0.25$
(d) $0$