https://file.aiquickdraw.com/mathbot/file/17268098488195l9hsuv6.jpg

The question in the image asks about probabilities based on the uniform distribution of a random variable \( X \) over the interval [0, 2].

The uniform distribution over [0, 2] means that the probability density function (PDF) is constant across this interval. Specifically, the PDF is \( f(x) = \frac{1}{2} \) for \( 0 \leq x \leq 2 \).

Now let's solve each probability:

### 1. \( P(X > 1.46) \)
This is the probability that \( X \) takes a value greater than 1.46. For a uniform distribution, this probability is simply the fraction of the remaining interval after 1.46:

\[
P(X > 1.46) = \frac{2 - 1.46}{2} = \frac{0.54}{2} = 0.27
\]

### 2. \( P(X < 0.08) \)
This is the probability that \( X \) takes a value less than 0.08. Again, for a uniform distribution, this probability is proportional to the length of the interval from 0 to 0.08:

\[
P(X < 0.08) = \frac{0.08 - 0}{2} = \frac{0.08}{2} = 0.04
\]

Therefore, the answers are:

- \( P(X > 1.46) = 0.27 \)
- \( P(X < 0.08) = 0.04 \)

Let me know if you'd like further details or explanations!

### Related questions:
1. How is the probability density function (PDF) defined for a uniform distribution?
2. How do you calculate probabilities for other types of continuous distributions (e.g., normal distribution)?
3. What is the difference between a cumulative distribution function (CDF) and a probability density function (PDF)?
4. Can uniform distributions be defined on infinite intervals, or are they always finite?
5. How do you find the mean and variance of a uniform distribution?

**Tip:** For uniform distributions, calculating probabilities always involves simple proportions of the interval length.

From the figure, what is P(X > 1.46) or P(X < 0.08) for a uniform distribution of a random variable X over [0, 2]?

This page covers how to solve uniform distribution probability problems, focusing on calculating P(X > 1.46) and P(X < 0.08) for a random variable X over the interval [0, 2]. Suitable for introductory statistics courses.

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