\( ( x ) = \left( \begin{array} { c } { n } \\ { x } \end{array} \right) p ^ { x } ( 1 - p ) ^ { n - x } , x = 0,1 , \ldots , n \)

The expression you've provided represents the probability mass function (PMF) of a binomial distribution. In this context: - \( n \) is the number of trials. - \( x \) is the number of successful trials (where \( x \) can take values from 0 to \( n \)). - \( p \) is the probability of success on each trial. - \( ( n \choose x ) \) is the binomial coefficient, which calculates the number of ways to choose \( x \) successes from \( n \) trials. The full expression can be rewritten as: \[ P(X = x) = \binom{n}{x} p^x (1 - p)^{n - x} \] where \( P(X = x) \) is the probability of getting exactly \( x \) successes in \( n \) independent Bernoulli trials. ### Key Properties of the Binomial Distribution: 1. **Mean**: The expected value (mean) of a binomial distribution is given by: \[ E(X) = n \cdot p \] 2. **Variance**: The variance of a binomial distribution is given by: \[ \text{Var}(X) = n \cdot p \cdot (1 - p) \] 3. **Standard Deviation**: The standard deviation is the square root of the variance: \[ \sigma = \sqrt{n \cdot p \cdot (1 - p)} \] 4. **Cumulative Distribution Function (CDF)**: The CDF of a binomial distribution gives the probability that the random variable \( X \) is less than or equal to a certain value \( k \): \[ P(X \leq k) = \sum_{x=0}^{k} \binom{n}{x} p^x (1 - p)^{n - x} \] This distribution is widely used in statistics, especially in scenarios involving binary outcomes (success/failure, yes/no, etc.). If you have any specific questions or need further clarification on this topic, feel free to ask!