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A **discrete random variable** is a type of random variable that can take on a countable number of distinct values. It is especially useful when outcomes are specific and distinct, like rolling a die or counting the number of full steps in an interval. In our exercise, variable \( X \) derived from \( U \) is a discrete random variable.Since \( U \) is from the continuous interval \([0, 1]\), \( X = [nU] \) transforms this into distinct integer values. Discretization happens because \( [\cdot] \) denotes the "floor function," which maps a real number to its greatest lesser integer. Hence, for each unique segment of \( U \), \( X \) will take an integer from \( 0 \) to \( n-1 \). Understanding discrete random variables helps in situations where outcomes can be counted, allowing for probability models and practical applications like simulations and statistical predictions.

**Probability calculation** is a critical step when dealing with random variables, helping to quantify the likelihood of different outcomes. Our task is to find the probability distribution of our new variable \( X = [nU] \).Given that \( U \) is uniformly distributed over \([0, 1]\), any sub-interval of equal length within \([0, 1]\) will also have equal probability. For each integer \( k \), the probability that \( X = k \) corresponds to the interval \( \frac{k}{n} \leq U < \frac{k+1}{n} \). Since these intervals all share the same length \( \frac{1}{n} \), each possible \( X \) value from \( 0 \) to \( n-1 \) has equal probability:- \( P(X = k) = \frac{1}{n} \) for \( k \in \{0, 1, \ldots, n-1\} \).This even distribution across these values makes \( X \) uniformly distributed, which is straightforward due to the nature of our discretization from \( U \). This is a simple yet powerful concept, allowing easy prediction and understanding of \( X \).

The **transformation of variables** is a technique used to map one set of values into another. This is particularly important when converting a continuous random variable into a discrete one. In our exercise, we transformed \( U \) into \( X = [nU] \).This transformation involves two operations:

• Scaling: Multiply \( U \) by \( n \) to scale up the uniform distribution over \([0, 1]\) into a range \([0, n)\).
• Floor Function: Convert the scaled continuous values into discrete integers with \( [nU] \).

This transformation neatly divides \([0, 1]\) into \( n \) parts, leading to the step-function behavior characteristic of discrete distributions. Each segment represents an equal span of \( U \), ensuring that results remain uniformly distributed by integer values.Understanding transformations bridges gaps between different types of variables, making it easier to model real-world events with mathematical precision.