STA301 Final Term Short Notes by Junaid

 Chebychev’s Inequality:

If X is a random variable having mean µ and variance σ2 > 0, and k is any positive constant, then the probability that a value of X falls within k standard deviations of the mean is at least.

 CONTINUOUS RANDOM VARIABLE:

A random variable X is defined to be continuous if it can assume every possible value in an interval [a, b], a < b, where a and b may be –∞ and +∞ respectively.

 JOINT DISTRIBUTIONS:

The distribution of two or more random variables which are observed simultaneously when an experiment is performed is called their JOINT

 BIVARIATE PROBABILITY FUNCTION:

The joint or bivariate probability distribution consisting of all pairs of values (xi, yj).

 MARGINAL PROBABILITY FUNCTIONS:

The point to be understood here is that, from the joint probability function for (X, Y), we can obtain the INDIVIDUAL probability function of X and Y. Such individual probability functions are called MARGINAL probability functions.

 CONDITIONAL PROBABILITY FUNCTION:

Let X and Y be two discrete r.v.’s with joint probability function f(x, y). Then the conditional probability function for X given Y = y, denoted as f(x|y).

 INDEPENDENCE:

Two discrete r.v.’s X and Y are said to be statistically independent, if and only if, for all possible pairs of values (xi, yj) the joint probability function f(x, y) can be expressed as the product of the two marginal probability functions.

 COVARIANCE OF TWO RANDOM VARIABLES:

The covariance of two r.v.’s X and Y is a numerical measure of the extent to which their values tend to increase or decrease together. It is denoted by σXY or Cov (X, Y), and is defined as the expected value of the product.

 BINOMIAL DISTRIBUTION:

The binomial distribution is a very important discrete probability distribution. It was discovered by James Bernoulli about the year 1700.

 PROPERTIES OF A BINOMIAL EXPERIMENT:

• Every trial results in a success or a failure.
• The successive trials are independent.
• The probability of success, p, remains constant from trial to trial.
• The number of trials, n, is fixed in advanced.

 Binomial probability distribution:

A probability distribution showing the probability of x successes in n trials of a binomial experiment.

 Binomial probability function:

The function used to compute probabilities in a binomial experiment.

 Binomial experiment:

A probability experiment having the following four properties: consists of n identical trials, two outcomes (success and failure) are possible on each trial, probability of success does not change from trial to trail, and the trials are independent.

 PROPERTIES OF A BINOMIAL EXPERIMENT:

• Every item selected will either be defective (i.e. success) or not defective (i.e. failure)
• Every item drawn is independent of every other item
• The probability of obtaining a defective item i.e. 7% is the same (constant) for all items. (This probability figure is according to relative frequency definition of probability.

 Hyper Geometric Probability function:

The function used to compute the probability of x successes in n trials when the trials are dependent.

 PROPERTIES OF HYPERGEOMETRIC EXPERIMENT:

•  The outcomes of each trial may be classified into one of two categories, success and failure.
•  The probability of success changes on each trial.
•  The successive trials are not independent.
•  The experiment is repeated a fixed number of times.

 Hyper Geometric Distribution:

There are many experiments in which the condition of independence is violated and the probability of success does not remain constant for all trials. Such experiments are called hyper geometric experiments.

 PROPERTIES OF THE HYPERGEOMETRIC

• If N becomes indefinitely large, the hyper geometric probability distribution tends to the BINOMIAL probability distribution.
• The above property will be best understood with reference to the following important points:
• There are two ways of drawing a sample from a population, sampling with replacement, and sampling without replacement.
• Also, a sample can be drawn from either a finite population or an infinite population.

 Poisson Probability Distribution:

A probability distribution showing the probability of x occurrences of an event over a specified interval of time or space.

 Poisson Probability Function:

The function used to compute Poisson probabilities.