Consider a uniform distribution from aequals3 to bequals26.

​(a) Find the probability that x lies between 4 and 18.

​(b) Find the probability that x lies between 5 and 11.

​(c) Find the probability that x lies between 6 and 15.

​(d) Find the probability that x lies between 8 and 19.

A uniform distribution is defined by a constant probability density function (PDF) between two endpoints, \( a \) and \( b \). For a continuous uniform distribution, the probability that \( x \) lies between two values \( x_1 \) and \( x_2 \) (where \( a \leq x_1 < x_2 \leq b \)) is calculated as the ratio of the interval length to the total length of the distribution.

Given the uniform distribution from \( a = 3 \) to \( b = 26 \), the PDF is:

\[
f(x) = \frac{1}{b - a} = \frac{1}{26 - 3} = \frac{1}{23}
\]

The probability that \( x \) lies between two values \( x_1 \) and \( x_2 \) is given by:

\[
P(x_1 \leq x \leq x_2) = \frac{x_2 - x_1}{b - a}
\]

Let's calculate the probabilities for each part.

### (a) Probability that \( x \) lies between 4 and 18:

\[
P(4 \leq x \leq 18) = \frac{18 - 4}{26 - 3} = \frac{14}{23}
\]

### (b) Probability that \( x \) lies between 5 and 11:

\[
P(5 \leq x \leq 11) = \frac{11 - 5}{26 - 3} = \frac{6}{23}
\]

### (c) Probability that \( x \) lies between 6 and 15:

\[
P(6 \leq x \leq 15) = \frac{15 - 6}{26 - 3} = \frac{9}{23}
\]

### (d) Probability that \( x \) lies between 8 and 19:

\[
P(8 \leq x \leq 19) = \frac{19 - 8}{26 - 3} = \frac{11}{23}
\]

### Summary of Results:
- (a) \( P(4 \leq x \leq 18) = \frac{14}{23} \)
- (b) \( P(5 \leq x \leq 11) = \frac{6}{23} \)
- (c) \( P(6 \leq x \leq 15) = \frac{9}{23} \)
- (d) \( P(8 \leq x \leq 19) = \frac{11}{23} \)

Would you like more details or explanations on these results?

Here are 5 related questions to expand your understanding:
1. How does the uniform distribution differ from a normal distribution?
2. What is the cumulative distribution function (CDF) for a uniform distribution?
3. How do you calculate the expected value for a uniform distribution?
4. What happens if the interval \([a, b]\) changes in a uniform distribution?
5. How can uniform distribution be applied in real-life scenarios like random number generation?

**Tip:** When dealing with probabilities in uniform distributions, remember that the area under the curve is always evenly distributed, so calculating probabilities is just about finding the proportion of the interval.

Consider a uniform distribution from a equals 3 to b equals 26.
(a) Find the probability that x lies between 4 and 18.
(b) Find the probability that x lies between 5 and 11.
(c) Find the probability that x lies between 6 and 15.
(d) Find the probability that x lies between 8 and 19.

Learn how to calculate probabilities for a uniform distribution from a = 3 to b = 26. This problem demonstrates the application of the uniform distribution formula to find probabilities for intervals between 4 and 19. Suitable for high school students in Grades 10-12.

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