NURSING

CHAPTER ONE 1.1. RANDOM VARIABLES One of the basic ideas in probability is that of a random variable. In many cases this is simply the numerical variable under consideration. For example, if a coin is tossed twice, the number of heads which turn up can be either 0, 1 or 2 according to the outcome of the experiment. We say that the number of heads is a random variable, since it expresses the result of the experiment as a number. Other simple random variables are the temperature of a chemical pr...

CHAPTER ONE 1.1. RANDOM VARIABLES One of the basic ideas in probability is that of a random variable. In many cases this is simply the numerical variable under consideration. For example, if a coin is tossed twice, the number of heads which turn up can be either 0, 1 or 2 according to the outcome of the experiment. We say that the number of heads is a random variable, since it expresses the result of the experiment as a number. Other simple random variables are the temperature of a chemical pr...

CHAPTER ONE 1.1. RANDOM VARIABLES One of the basic ideas in probability is that of a random variable. In many cases this is simply the numerical variable under consideration. For example, if a coin is tossed twice, the number of heads which turn up can be either 0, 1 or 2 according to the outcome of the experiment. We say that the number of heads is a random variable, since it expresses the result of the experiment as a number. Other simple random variables are the temperature of a chemical pr...

CHAPTER ONE 1.1. RANDOM VARIABLES One of the basic ideas in probability is that of a random variable. In many cases this is simply the numerical variable under consideration. For example, if a coin is tossed twice, the number of heads which turn up can be either 0, 1 or 2 according to the outcome of the experiment. We say that the number of heads is a random variable, since it expresses the result of the experiment as a number. Other simple random variables are the temperature of a chemical pr...

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 referred to as a beta random variable if and only if its probability density function is given by where are the parameters of the distribution. In recent years, the beta distribution has found ...

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, in situations where we have say a random variable X representing the number of trials required before the first success, then X has a geometric distribution. The Mean, Variance and Moment generating...

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; Hence f (x) is a p. d. f. A random variable X that has a p.d.f. of the form is said to have a binomial distribution, and any such is called a binomial p.d.f. A binomial distribut...

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 particular distributions but can take on different values for different members of families of distributions of the same kind. The most common parameters are the lower moments, mainly and . 1.1.1. ...

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;   Example 2.8 Consider the random variable X whose p.d.f. is where is the parameter of the distribution. (a) Obtain the moment generating function of X. (b) Hence obtain the mean and variance of X...