Question

Find the following probabilities, if X is an uniform variable whose density function is shown below. Write as a decimal.

P(X<45) =

P(12<X<32) =

Answer 1

To find the probabilities for a uniform random variable \( X \), we need to know the range of the uniform distribution.

1. The cumulative distribution function (CDF) gives the probability \( P(X < x) \). For a uniform distribution defined on the interval \([a, b]\), the CDF is given by:

\[
P(X < x) = \frac{x - a}{b - a} \quad \text{for } a \leq x \leq b
\]

2. The probability of \( P(a < X < b) \) can be calculated as:

\[
P(a < X < b) = \frac{b - a}{d - c}
\]

where \( [c, d] \) is the interval of the uniform variable.

Without the specific range of \( X \) provided, let's assume \( X \) ranges from \( a \) to \( b \).

### Assumptions:
Let's assume \( X \) is uniformly distributed in the interval \([10, 50]\).

Now, let's calculate the probabilities based on these assumptions:

1. **For \( P(X < 45) \)**:
\[
P(X < 45) = \frac{45 - 10}{50 - 10} = \frac{35}{40} = 0.875
\]

2. **For \( P(12 < X < 32) \)**:
\[
P(12 < X < 32) = \frac{32 - 12}{50 - 10} = \frac{20}{40} = 0.5
\]

### Summary:
- \( P(X < 45) = 0.875 \)
- \( P(12 < X < 32) = 0.5 \)

Please provide the specific range of the uniform variable if different from this assumption for accurate calculations.