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Statistical Analysis. Pedro Flores. Conditional Probability. The conditional probability of an event B is the probability that the event will occur given the knowledge that an event A has already occurred. This probability is written P(B|A).
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Statistical Analysis Pedro Flores
Conditional Probability • The conditional probability of an event B is the probability that the event will occur given the knowledge that an event A has already occurred. • This probability is written P(B|A). • If events A and B are not independent, then the probability of the intersection of A and B (the probability that both events occur) is defined by P(A and B) = P(A)P(B|A). • From this definition, the conditional probability P(B|A) is easily obtained by dividing by P(A):
Standard Deviation • Lower case sigma means 'standard deviation'.Capital sigma means 'the sum of'.x bar means 'the mean' • The standard deviation measures the spread of the data about the mean value. It is useful in comparing sets of data which may have the same mean but a different range.
Confidence Levels • The confidence level is the probability value associated with a confidence interval. • It is often expressed as a percentage. For example, say , then confidence level = (1-0.05) = 0.95, that is, a 95% confidence level. *********************************************************** • Example • Suppose an opinion poll predicted that, if the election were held today, the Conservative party would win 60% of the vote. The pollster might attach a 95% confidence level to the interval 60% plus or minus 3%. That is, he thinks it very likely that the Conservative party would get between 57% and 63% of the total vote.
Normal Distributions Normal distributions model continuous random variables. A Normal random variable should be capable of assuming any value on the real line, though this requirement is often waived in practice. For example, height at a given age for a given gender in a given racial group is adequately described by a Normal random variable even though heights must be positive. A continuous random variable X, taking all real values in the range is said to follow a Normal distribution with parameters µ and if it has probability density function We write This probability density function (p.d.f.) is a symmetrical, bell-shaped curve, centered at its expected value µ. The variance is .
Poisson Distributions • Poisson distributions model discrete random variables. Typically, a Poisson random variable is a count of the number of events that occur in a certain time interval or spatial area. • A discrete random variable X is said to follow a Poisson distribution with parameter m, written X ~ Po(m), if it has probability distribution where x = 0, 1, 2, ..., n m > 0. • The following requirements must be met: the length of the observation period is fixed in advance; the events occur at a constant average rate; the number of events occurring in disjoint intervals are statistically independent. • The Poisson distribution has expected value E(X) = m and variance V(X) = m; i.e. E(X) = V(X) = m.
Binomial Distributions • Binomial distributions model discrete random variables. Typically, a binomial random variable is the number of successes in a series of trials. • A discrete random variable X is said to follow a Binomial distribution with parameters n and p if it has probability distribution • where x = 0, 1, 2, ......., n n = 1, 2, 3, ....... p = success probability; 0 < p < 1 • The trials must meet the following requirements: • the total number of trials is fixed in advance; • there are just two outcomes of each trial; success and failure; • the outcomes of all the trials are statistically independent; • all the trials have the same probability of success. • The Binomial distribution has expected value E(X) = np and variance V(X) = np(1-p).
