Let's go through each of these problems step-by-step:

Question 22:

Given: A random variable $X$ follows a uniform distribution between 10 and 18. We are asked to find the probability that $X$ is between 12 and 14.

For a uniform distribution, the probability of any event within an interval is proportional to the length of the interval relative to the total length of the distribution.

• The total length of the interval $[10, 18]$ is: $18 - 10 = 8$
• The length of the interval $[12, 14]$ is: $14 - 12 = 2$

The probability is given by the ratio of these two lengths: $P(12 \leq X \leq 14) = \frac{\text{Length of desired interval}}{\text{Length of total interval}} = \frac{2}{8} = 0.25$

Thus, the correct answer is A) 0.25.

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Question 23:

Given: A random variable $X$ is uniformly distributed over the interval $[0, 8]$. We are asked to find the probability that $X$ is less than 6.

• The total length of the interval $[0, 8]$ is: $8 - 0 = 8$
• The length of the interval $[0, 6]$ is: $6 - 0 = 6$

Again, the probability is the ratio of these lengths: $P(X < 6) = \frac{6}{8} = 0.75$

Thus, the correct answer is A) 0.75.

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Question 24:

Given: A random variable $X$ follows a uniform distribution between 2 and 6. We are asked to find the expected value (mean) of $X$.

For a uniform distribution between $a$ and $b$, the expected value is given by: $E(X) = \frac{a + b}{2}$ Here, $a = 2$ and $b = 6$: $E(X) = \frac{2 + 6}{2} = \frac{8}{2} = 4$

Thus, the correct answer is C) 4.

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Let me know if you'd like further explanation on any of these or if you have additional questions!

Related Questions:

1. What is the probability that $X$ is between 15 and 17 for a uniform distribution between 10 and 18?
2. How would the probability change if the interval for $X$ was $[0, 10]$ instead of $[0, 8]$?
3. How do you calculate the variance for a uniform distribution?
4. If $X$ is uniformly distributed between 1 and 5, what is the probability that $X$ is greater than 3?
5. How would the expected value formula change for a uniform distribution over a different interval?

Tip:

For uniform distributions, the probability of any subinterval is directly proportional to the size of the subinterval relative to the total interval. Keep this in mind for quick calculations!