UPPSC GOVERNMENT DEGREE COLLEGE COMMERCE UNIT-IV

Binomial Distribution The binomial distribution is a probability distribution that summarizes the likelihood that a value will take one of two independent values under a given set of parameters or assumptions. The underlying assumptions of the binomial distribution are that there is only one outcome for each trial, that each trial has the same probability of success, and that each trial is mutually exclusive, or independent of each other. The binomial distribution is a common discrete distribution used in statistics, as opposed to a continuous distribution, such as the normal distribution. This is because the binomial distribution only counts two states, typically represented as 1 (for a success) or 0 (for a failure) given a number of trials in the data. The binomial distribution, therefore, represents the probability for x successes in n trials, given a success probability p for each trial. The mean of the binomial distribution is np, and the variance of the binomial distribution is np (1 − p). When p = 0.5, the distribution is symmetric around the mean. When p > 0.5, the distribution is skewed to the left. When p < 0.5, the distribution is skewed to the right. The binomial distribution is the sum of a series of multiple independent and identically distributed Bernoulli trials. In a Bernoulli trial, the experiment is said to be random and can only have two possible outcomes: success or failure. 1. In a Binomial Distribution, if ‘n’ is the number of trials and ‘p’ is the probability of success, then the mean value is given by np 2. In a Binomial Distribution, if p, q and n are probability of success, failure and number of trials respectively then variance is given by npq Explanation: The variance (V) for a Binomial Distribution is given by V = npq Standard Deviation = √variance √.npq 3. If ‘X’ is a random variable, taking values ‘x’, probability of success and failure being ‘p’ and ‘q’ respectively and ‘n’ trials being conducted, then what is the probability that ‘X’ takes values ‘x’? Use Binomial Distribution P(X = x) = nCx px q(n-x) 4. If ‘p’, ‘q’ and ‘n’ are probability pf success, failure and number of trials respectively in a Binomial Distribution, what is its Standard Deviation? √ npq 5. It is suitable to use Binomial Distribution only for -Small values of ‘n’ 7. For larger values of ‘n’, Binomial Distribution -tends to Poisson Distribution 8. Binomial Distribution, if p = q, then P(X = x) is given by? nCx (0.5)n 9. Binomial Distribution is a ___________ Discrete distribution 10. The binomial distribution is a discrete distribution. A binomial experiment has the following properties: experiment consists of n identical and independent trials each trial results in one of two outcomes: success or failure P(success) = p P(failure) = q = 1 - p for all trials The random variable of interest, X, is the number of successes in the n trials. X has a binomial distribution with parameters n and p The mean (expected value) of a binomial random variable is The standard deviation of a binomial random variable is µ = np σ = npq POISON DISTRIBUTION 1. In a Poisson Distribution, if ‘n’ is the number of trials and ‘p’ is the probability of success, then the mean value is given by? m = np 2. If ‘m’ is the mean of a Poisson Distribution, then variance is given by m 3. The p.d.f of Poisson Distribution is given by ___________ a) e−mmxx! 4. In a Poisson Distribution, the mean and variance are equal.. 5. Poisson distribution is applied for Discrete Random Variable 6. If ‘m’ is the mean of Poisson Distribution, the P(0) is given by ___________a) e-m NORMAL DISTRIBUTION In probability theory, a normal (or Gaussian or Gauss or Laplace–Gauss) distribution is a type of continuous probability distribution for a real-valued random variable. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. The simplest case of a normal distribution is known as the standard normal distribution. This is a special case when and , and it is described by this probability density function The normal distribution is a subclass of the elliptical distributions. • The area under the curve and over the -axis is unity (i.e. equal to one). Properties of a normal distribution • The mean, mode and median are all equal. • The curve is symmetric at the center (i.e. around the mean, μ). • Exactly half of the values are to the left of center and exactly half the values are to the right. • The total area under the curve is 1. The Standard Normal Model A standard normal model is a normal distribution with a mean of 0 and a standard deviation of 1. The normal distribution is symmetric and has a skewness of zero. If the distribution of a data set has a skewness less than zero, or negative skewness, then the left tail of the distribution is longer than the right tail; positive skewness implies that the right tail of the distribution is longer than the left. The normal distribution has a kurtosis of three, which indicates the distribution has neither fat nor thin tails. the interval [μ ± 1σ] corresponds to P = 0.683, the interval [μ ± 2σ] corresponds to P = 0.954, and the interval [μ ± 3σ] corresponds to P = 0.997 Most statisticians give credit to French scientist Abraham de Moivre for the discovery of normal distributions. In the second edition of “The Doctrine of Chances,”