Question

Given x as a binomial random variable resulting from the performance of n Bernoulli trials with a probability of success p, determine the expected value and variance of x.

Answer 1

The expected value
`(\(E[x]\))` of a binomial random variable x resulting from
`\(n\)` Bernoulli trials with a probability of success p is
`\(E[x] = np\)`. The variance
`(\(Var[x]\))` is given by
`\(Var[x] = np(1-p)\).`

Step-by-step explanation:

In a binomial distribution, where \(x\) represents the number of successes in \(n\) independent Bernoulli trials, the expected value is a measure of the center of the distribution, and the variance indicates the spread or dispersion.

1. Expected Value
`(\(E[x]\))`:

The expected value is calculated by multiplying the number of trials
`(\(n\))`by the probability of success
`(\(p\))`. Mathematically,
`\(E[x] = np\)`. This formula represents the average or mean number of successes one would expect in \(n\) trials.

2. Variance
`(\(Var[x]\))`:

The variance of a binomial distribution is found by multiplying the number of trials n by the probability of success p and the probability of failure
`(\(1-p\))` . The formula is
`\(Var[x] = np(1-p)\)`. This quantifies the spread or variability in the distribution.

Understanding these measures is crucial for characterizing the behavior of a binomial random variable. They provide insights into the central tendency and the extent to which individual trials deviate from the expected value.