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
$\endgroup$ 12 Answers
$\begingroup$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 $P(X=4.75)=0.$
$\endgroup$ 1 $\begingroup$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.
