Mean and variance of binomial distribution are
np, npq
Binomial Distribution Concepts
The binomial distribution is a fundamental concept in probability and statistics. It describes the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes: success or failure. The probability of success remains the same for every trial.
Key parameters for a binomial distribution include:
- Number of Trials (\(n\)): This represents the total fixed number of times an experiment or process is repeated.
- Probability of Success (\(p\)): This is the constant probability of achieving a 'success' in any single trial.
- Probability of Failure (\(q\)): This is the probability of not achieving a 'success' (i.e., a 'failure') in any single trial. It is calculated as \(q = 1 - p\).
Mean of Binomial Distribution
The mean of a binomial distribution, often referred to as the expected value (\(E(X)\) or \(\mu\)), is a measure of central tendency. It tells us the average number of successes we can expect to observe over many repetitions of the experiment. For a binomial distribution, the mean is straightforward to calculate.
The formula for the mean of a binomial distribution is:
\[ \text{Mean} = np \]
Where:
- \(n\) is the total number of trials.
- \(p\) is the probability of success in a single trial.
For example, if you flip a fair coin 10 times (\(n=10\)), the probability of getting heads is \(0.5\) (\(p=0.5\)). The expected number of heads (mean) would be \(10 \times 0.5 = 5\).
Variance of Binomial Distribution
The variance of a binomial distribution, denoted as \(Var(X)\) or \(\sigma^2\), quantifies the spread or dispersion of the distribution. It indicates how much the number of successes is likely to deviate from the mean. A higher variance means the outcomes are more spread out, while a lower variance means they are more clustered around the mean.
The formula for the variance of a binomial distribution is:
\[ \text{Variance} = npq \]
Where:
- \(n\) is the total number of trials.
- \(p\) is the probability of success in a single trial.
- \(q\) is the probability of failure in a single trial (\(q = 1 - p\)).
Following the coin flip example (\(n=10, p=0.5\)), the probability of failure \(q\) would also be \(1 - 0.5 = 0.5\). The variance would be \(10 \times 0.5 \times 0.5 = 2.5\).
Summarizing Binomial Distribution Measures
The table below summarizes the key statistical measures for a binomial distribution:
| Measure | Formula |
|---|---|
| Mean (Expected Value) | \(np\) |
| Variance | \(npq\) |
| Standard Deviation | \(\sqrt{npq}\) |
Therefore, the mean and variance of a binomial distribution are \(np\) and \(npq\), respectively.
Important Questions from Statistical Variables
The formula to calculate the coefficient of quartile deviation is
Let X be a normal random variable with mean zero and variance 9. If a = P(X ≥ 3) then P(|X| ≤ 3) equals:
Let X be a Poisson random variable such that 2P(X = 0) = P(X = 2). Then the standard deviation of X is:
The median of 7, 5, 8, x, 12, 17 is 10, then what is the value of x?
Find the median if the given data set is:
3, 3, 7, 8, 12, 13, 16, 19
