Probability and Statistics II

There are many problems in which it is of interest to know the probability that the value of a random variable is less than or equal to some real number x. Thus, let us write the probability that X takes on a value less than or equal to x as and refer to this function defined for all real numbers...

As in the discrete case, there are many problems in which it is of interest to know the probability that the value of a continuous random variable X is less than or equal to some real number x. Thus, let us make the following definition analogous to Definition 1.3. Definition 1.6 (Distribution Fu...

Properties of Expectation Let X be a random variable and c a constant. Then we have: (i) E (c) = c, for c a constant (ii) E [c g (x)] = c E [g (x)], where c is a constant (iii) E [c1 g1 (x) + c2 g2 (x)] = c1 E [g1 (x)] + c2 E [g2 (x)] where c1 and c2 are constants. Proof We shall assume th...

MOMENT GENERATING FUNCTION (m.g.f) The moment generating function (m.g.f) of a random variable X is defined as: is the expected value of . We are required to show that: and Proof We know that is differentiable. when t = 0; when t = 0; But we know that;   Ex...

CHAPTER THREE: UNIVARIATE DISTRIBUTIONS 1.1. DISCRETE DISTRIBUTIONS In this chapter we shall study some of the probability distributions that figure most prominently in statistical theory and applications. We shall also study their parameters, that is, the quantities that are constants for part...

Example 3.1 Given , p = probability of success and q is the probability of failure. Show that f (x) is a discrete probability density function (p. d. f). Solution To show that a function is a discrete p. d. f. we have to show that its sum is unity, i.e., But from the binomial formula;...

1.1.1. GEOMETRIC (PASCAL) DISTRIBUTION A random variable X is defined to have a geometric distribution, if the discrete probability distribution function (p.d.f) of X is given by 0<p<1. p = probability of success and q is the probability of failure. This distribution occurs in real life,...

THE BETA DISTRIBUTION X is uniform on the interval [a, b], the p.d.f. of the uniform distribution is defined as: The uniform density is a special case of the beta distribution, which is defined in the following way: Definition 3.2.5. A random variable X has a beta distribution and it is ref...

3.1. UNIFORM DISTRIBUTIONS: SPECIAL PROBABILITY DENSITY FUNCTIONS In this section we shall study some of the probability densities that figure most prominently is statistical theory and in applications. 3.2.1. THE UNIFORM DISTRIBUTION Definition 3.2.1. A random variable X has a uniform dist...

From Equation (3.2.3), we see that the normal distribution has the cumulative distribution function , ……………………………………………(3.2.4) From this we obtain , ………………………………(3.2.5) The integral in (3.2.4) cannot be evaluated by elementary method...

CHAPTER FOUR: BIVARIATE DISTRIBUTION THEORY We earlier defined random variable as a real-valued function over a sample space with a probability measure, and it stands to reason that many different random variables can be defined over one and the same sample space. In this section we shall be conce...

MARGINAL DISTRIBUTIONS Definition 5 If X and Y are discrete random variables and is the value of their joint probability distribution at , the function given by: for each x within the range of X is called the marginal probability distribution function of X. Correspondingly the function given ...

BIVARIATE EXPECTATION There are many problems in which we are interested not only in the expected value of a random variable X, but also in the expected values of random variables X and Y. Theorem 1 If X and Y are discrete random variables and is the value of their joint probability distributio...