Let Y be a random number between 0 and 5. So Y has uniform distribution on the interval [0,5]. (a) What is the height of the uniform density curve U([0,5])? (b) With 20% chance Y is bigger than which value? (c) What is the chance that Y is between 0 and 2.5 or between 4 and 5?
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
![### Uniform Distribution Problem Set
Let \( Y \) be a random number between 0 and 5. So \( Y \) has uniform distribution on the interval \([0, 5]\).
1. **What is the height of the uniform density curve \( U([0, 5]) \)?**
_Answer box: ___________
2. **With 20% chance \( Y \) is bigger than which value?**
_Answer box: ___________
3. **What is the chance that \( Y \) is between 0 and 2.5 or between 4 and 5?**
_Answer box: ___________
### Explanation
#### Uniform Distribution Overview
A uniform distribution, in this case, means every number between 0 and 5 is equally likely to be chosen. The height of the uniform density curve can be found by considering that the total area under the curve must equal 1. Therefore, the height \( h \) times the width (which is 5 - 0 = 5) must equal 1, resulting in \( h = \frac{1}{5} \).
### Answer Key
1. **Height of the Uniform Density Curve:**
\[ \frac{1}{5} \]
2. **20% Chance \( Y \) is Bigger Than:**
For \( Y \) to have a 20% chance of being bigger than a specific value \( x \), we need to find where the area under the curve from \( x \) to 5 equals 0.20. Use the formula for the cumulative distribution function (CDF) of a uniform distribution to find \( x \):
\[ P(Y > x) = 0.20 \]
\[ 1 - \frac{x}{5} = 0.20 \]
Solve for \( x \):
\[ \frac{x}{5} = 0.80 \]
\[ x = 4 \]
3. **Chance that \( Y \) is Between 0 and 2.5 or Between 4 and 5:**
The probability is found by summing the probabilities of the two intervals:
\[ P(0 \leq Y \leq 2.5) + P(4 \leq Y \leq 5) \]
\[ \frac{2.5 - 0}{5](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd7856584-59fd-4634-ab45-c4c5f6444c59%2F1cef9640-54cc-4632-a372-51ba568a168c%2F4a59137_processed.png&w=3840&q=75)
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