User:Aganze Mwaza Eric/sandbox

Who is Aganze Mwaza Eric? Aganze Mwaza Eric is a currently a computer science student at Strathmore University who has done his high school in 2018 at EDAP-ISP Bukavu in DRC. He has gotten a scholarship of strathmore University through a mathematical context in February 2018 in Bukavu. His passion is to become an ethical hacker.

Random variables by Eric Mwaza
The probabilities associated with each outcome are described by the following table:

Outcome 	1	2	3	4 Probability	0.1	0.3	0.4	0.2

The probability that X is equal to 2 or 3 is the sum of the two probabilities: P(X = 2 or X = 3) = P(X = 2) + P(X = 3) = 0.3 + 0.4 = 0.7. Similarly, the probability that X is greater than 1 is equal to 1 - P(X = 1) = 1 - 0.1 = 0.9, by the complement rule.

This distribution may also be described by the probability histogram shown to the right: All random variables (discrete and continuous) have a cumulative distribution function. It is a function giving the probability that the random variable X is less than or equal to x, for every value x. For a discrete random variable, the cumulative distribution function is found by summing up the probabilities.

(Definition taken from Valerie J. Easton and John H. McColl's Statistics Glossary v1.1) Example The cumulative distribution function for the above probability distribution is calculated as follows: The probability that X is less than or equal to 1 is 0.1, the probability that X is less than or equal to 2 is 0.1+0.3 = 0.4, the probability that X is less than or equal to 3 is 0.1+0.3+0.4 = 0.8, and the probability that X is less than or equal to 4 is 0.1+0.3+0.4+0.2 = 1.

The probability histogram for the cumulative distribution of this random variable is shown to the right: Continuous Random Variables A continuous random variable is one which takes an infinite number of possible values. Continuous random variables are usually measurements. Examples include height, weight, the amount of sugar in an orange, the time required to run a mile.

(Definition taken from Valerie J. Easton and John H. McColl's Statistics Glossary v1.1)

A continuous random variable is not defined at specific values. Instead, it is defined over an interval of values, and is represented by the area under a curve (in advanced mathematics, this is known as an integral). The probability of observing any single value is equal to 0, since the number of values which may be assumed by the random variable is infinite.

Suppose a random variable X may take all values over an interval of real numbers. Then the probability that X is in the set of outcomes A, P(A), is defined to be the area above A and under a curve. The curve, which represents a function p(x), must satisfy the following:

1: The curve has no negative values (p(x) > 0 for all x)    2: The total area under the curve is equal to 1.

A curve meeting these requirements is known as a density curve.

The Uniform Distribution A random number generator acting over an interval of numbers (a,b) has a continuous distribution. Since any interval of numbers of equal width has an equal probability of being observed, the curve describing the distribution is a rectangle, with constant height across the interval and 0 height elsewhere. Since the area under the curve must be equal to 1, the length of the interval determines the height of the curve.

The following graphs plot the density curves for random number generators over the intervals (4,5) (top left), (2,6) (top right), (5,5.5) (lower left), and (3,5) (lower right). The distributions corresponding to these curves are known as uniform distributions. Consider the uniform random variable X defined on the interval (2,6). Since the interval has width = 4, the curve has height = 0.25 over the interval and 0 elsewhere. The probability that X is less than or equal to 5 is the area between 2 and 5, or (5-2)*0.25 = 0.75. The probability that X is greater than 3 but less than 4 is the area between 3 and 4, (4-3)*0.25 = 0.25. To find that probability that X is less than 3 or greater than 5, add the two probabilities: P(X < 3 and X > 5) = P(X < 3) + P(X > 5) = (3-2)*0.25 + (6-5)*0.25 = 0.25 + 0.25 = 0.5.

The uniform distribution is often used to simulate data. Suppose you would like to simulate data for 10 rolls of a regular 6-sided die. Using the MINITAB "RAND" command with the "UNIF" subcommand generates 10 numbers in the interval (0,6):

MTB > RAND 10 c2; SUBC> unif 0 6.

Assign the discrete random variable X to the values 1, 2, 3, 4, 5, or 6 as follows: if 05, X=6. Use the generated MINITAB data to assign X to a value for each roll of the die:

Uniform Data		X Value 4.53786			5 5.77474			6 3.69518			4 1.03929			2 4.23835			5 0.37096			1 0.75272			1 5.56563			6 0.89045			1 3.18086			4

Another type of continuous density curve is the normal distribution. The area under the curve is not easy to calculate for a normal random variable X with mean and standard deviation. However, tables (and computer functions) are available for the standard random variable Z, which is computed from X by subtracting and dividing by. All of the rules of probability apply to the normal distribution.