Cant Find the Height of this Uniform Probability Distribution The waiting times between a subway departure schedule and the arrival of a passenger are uniformly distributed between 0 and 8 minutes. Find the probability that a randomly selected passenger has a waiting time less than 4.75 minutes
For a density curve to be a graph of a continuous probability​ distribution, it must have a total area under the curve equal to 1 and the curve cannot fall below the​ x-axis
The length of the uniform distribution is the difference between the maximum and minimum values
In this​ situation, the length of the uniform distribution is given by the following equation.
8-0= 8 min
Since the uniform distribution is​ rectangular, has a length of 8 
and an area of​ 1, determine the height of the uniform​ distribution, rounding to two decimal places.
I am not sure how to determine the height for this question
 A: The general formula for the Uniform Probability Density Function is: $f(x)=\frac{1}{b-a}$, where $b-a$ is the given interval.
Hence, $f(x)=\frac{1}{8-0}=\frac{1}{8}.$
Now, in order to calculate the probability of waiting time less than 4.75 minutes, one must integrate the Uniform Probability Density Function in the required interval:
$$P(0 < X < 4.75)=\int_0^{4.75}f(x)dx=\int_0^{4.75} \frac{1}{8}dx=\frac{1}{8}x|_0^{4.75}=4.75 \cdot \frac{1}{8}=0.59375.$$
Also, notice that 
$P(0<X<4.75)=P(0\leq X \leq 4.75)$, which implies $P(X=4.75)=0.$
A: You have $X \sim \mathsf{Unif}(0,8)$ and you seek $P(x < 4.75).$
The density function for $X$ is of the form $f(x) = c,$ for $0 < x < 8,$
and $f(x) = 0,$ for all other $x.$ You need the area under $f(x)$ within
$(0,8)$ to be $1.$ So you have a rectangle with base $8$ and height $c,$
and you want $8c = 1.$ That implies $c = 1/8.$ 
Now to find $P(X < 4.75) = P(0 < X < 4.75)$ you want the area of a rectangle 
with base $4.75$ and height $1/8,$ so your answer is $4.75/8 = ??.$
In the figure below, you want the area within the rectangle to the left
of the vertical red dotted line.

