Continuous Uniform Distributions
Solving for X
Learning Objectives
Calculate the missing [latex]x[/latex] when given a probability in continuous uniform problems.
We can solve for [latex]x[/latex] when given a probability, [latex]p[/latex]:
- Use [latex]p = P(x_1 \le x \le x_2) = \frac{x_2 - x_1}{b-a}[/latex]
- And, use algebra to solve for the missing [latex]x[/latex] value.
Travel Times (at least ExERCISE)
Let us revisit the commute times example and calculate the [latex]x[/latex]-value when given the probability of at least [latex]x[/latex].
Example 36.1.1
Problem Setup: During the morning commute, the time it takes to drive to BCIT follows a uniform distribution and is between 30 and 55 minutes.
Question: 50% of the time it takes you at least how long to get to campus?
You Try: Let us first determine the values to input in the equation below:
\[ P(x_1 \le x \le x_2) = \frac{x_2-x_1}{b-a} = p \]
Solution: Now, use algebra to solve for the minimum time it takes you half of the time to get to campus. Select your answer from the options below:
Click below to reveal the full solutions for the above exercise.
First, the values we input are the following:
- [latex]a = 30[/latex] (minimum possible travel time)
- [latex]b = 55[/latex] (maximum possible travel time)
- [latex]x_1 = X[/latex] (we do not know this minimum [latex]x[/latex]-value)
- [latex]x_2 = 55[/latex] (we know the probability of at least a certain travel time – so there is no upper limit)
- [latex]p = 50\% = 0.5[/latex] (50% of the time it takes you at least how long?)
Let us plug them into the formula [latex]P(x_1 \le x \le x_2) = \frac{x_2-x_1}{b-a} = p[/latex] and solve for [latex]X[/latex]:
[latex]\begin{align} P(X \le x \le 55) = \frac{55-X}{55-30} &= 0.5 \\ 55-X &= 0.5 \times (55-30) \\ 55-X &= 0.5 \times 25 = 12.5 \\ -X &= 12.5-55 \\ -X &= -42.5 \\ X &= 42.5 \\ \end{align}[/latex]
Travel Times Written Example (Less Than Video)
Let us now change up the problem slightly and show the solution in a video (see below).
Example 36.1.2
Problem Setup: Let us continue on with the previous example:
- Your commute times follow a uniform distribution
- And are between 30 and 55 minutes.
Question: 30% of the time, it takes you less than how many minutes to get to campus? Solve the problem by hand.
Solution: Click here to download the written solutions. Also, see the video below:
Travel Times Using Excel’s Goal Seek (Video)
Let us finish up this section by going through how to use Excel’s Goal Seek to solve the previous example (see below).
Example 36.1.3
Problem Setup: Let us continue on with the previous example:
- Your commute times follow a uniform distribution
- And are between 30 and 55 minutes.
Question: 30% of the time, it takes you less than how many minutes to get to campus? Solve the problem using Excel’s Goal Seek.
Solution: Click here to download the Excel solutions shown in the solution video below:
Key Takeaways (EXERCISE)
Key Takeaways: Solving for X
Your Own Notes (EXERCISE)
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