To find the probabilities for a uniform random variable \( X \), we need to know the range of the uniform distribution.
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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:
\[ P(X < x) = \frac{x - a}{b - a} \quad \text{for } a \leq x \leq b \]
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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:
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For \( P(X < 45) \): \[ P(X < 45) = \frac{45 - 10}{50 - 10} = \frac{35}{40} = 0.875 \]
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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.
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