Moment Generating Functions of Random Variables

Moment Generating Functions of Random Variables One way to calculate the mean and variance of a probability distribution is to find the expected values of the random variables X and X2. We use the notation E(X) and E(X2) to denote these expected values. In general, it is difficult to calculate E(X) and E(X2) directly. To get around this difficulty, we use some more advanced mathematical theory and calculus. The end result is something that makes our calculations easier. The strategy for this problem is to define a new function, of a new variable t that is called the moment generating function. This function allows us to calculate moments by simply taking derivatives. Assumptions Before we define the moment generating function, we begin by setting the stage with notation and definitions. We let X be a discrete random variable. This random variable has the probability mass function f(x). The sample space that we are working with will be denoted by S. Rather than calculating the expected value of X, we want to calculate the expected value of an exponential function related to X. If there is a positive real number r such that E(etX) exists and is finite for all t in the interval [-r, r], then we can define the moment generating function of X. Definition The moment generating function is the expected value of the exponential function above. In other words, we say that the moment generating function of X is given by: M(t) E(etX) This expected value is the formula ÃŽ £ etx f (x), where the summation is taken over all x in the sample space S. This can be a finite or infinite sum, depending upon the sample space being used. Properties The moment generating function has many features that connect to other topics in probability and mathematical statistics. Some of its most important features include: The coefficient of etb is the probability that X b.Moment generating functions possess a uniqueness property. If the moment generating functions for two random variables match one another, then the probability mass functions must be the same. In other words, the random variables describe the same probability distribution.Moment generating functions can be used to calculate moments of X. Calculating Moments The last item in the list above explains the name of moment generating functions and also their usefulness. Some advanced mathematics says that under the conditions that we laid out, the derivative of any order of the function M (t) exists for when t 0. Furthermore, in this case, we can change the order of summation and differentiation with respect to t to obtain the following formulas (all summations are over the values of x in the sample space S): M’(t) ÃŽ £ xetx f (x)M’’(t) ÃŽ £ x2etx f (x)M’’’(t) ÃŽ £ x3etx f (x)M(n)’(t) ÃŽ £ xnetx f (x) If we set t 0 in the above formulas, then the etx term becomes e0 1. Thus we obtain formulas for the moments of the random variable X: M’(0) E(X)M’’(0) E(X2)M’’’(0) E(X3)M(n)(0) E(Xn) This means that if the moment generating function exists for a particular random variable, then we can find its mean and its variance in terms of derivatives of the moment generating function. The mean is M’(0), and the variance is M’’(0) – [M’(0)]2. Summary In summary, we had to wade into some pretty high-powered mathematics, so some things were glossed over. Although we must use calculus for the above, in the end, our mathematical work is typically easier than by calculating the moments directly from the definition.

Apartheid Era Pass Laws of South Africa

Apartheid Era Pass Laws of South Africa As a system, apartheid focused on separating South African Indian, Colored, and African citizens according to their race. This was done to promote the superiority of Whites and to establish the minority White regime. Legislative laws were passed to accomplish this, including the Land Act of 1913, the Mixed Marriages Act of 1949, and the Immorality Amendment Act of 1950- all of which were created to separate the races. Under apartheid, pass laws were designed to control the movement of Africans and they are considered one of the most grievous methods that the South African government used to support apartheid. The resulting legislation (specifically Abolition of Passes and Co-ordination of Documents Act No. 67 of 1952) introduced in South Africa required black Africans to carry identity documents in the form of a reference book when outside a set of reserves (later known as homelands or bantustans). Pass laws evolved from regulations that the Dutch and British enacted during the 18th and 19th-century slave economy of the Cape Colony. In the 19th century, new pass laws were enacted to ensure a steady supply of cheap African labor for the diamond and gold mines. In 1952, the government passed an even more stringent law that required all African men age of 16 and over to carry a reference book (replacing the previous passbook) which held their personal and employment information. (Attempts to force women to carry pass books in 1910, and again during the 1950s, caused strong protests.) Pass Book Contents The pass book was similar to a passport in that it contained details about the individual, including a photograph, fingerprint, address, the name of his employer, how long the person had been employed, and other identifying information. Employers often entered an evaluation of the pass holders behavior. As defined by law, an employer could only be a White person. The pass also documented when permission was requested to be in a certain region and for what purpose, and whether that request was denied or granted. Under law, any governmental employee could remove these entries, essentially removing permission to stay in the area. If a pass book didnt have a valid entry, officials could arrest its owner and put him in prison. Colloquially, passes were known as the dompas, which literally meant the dumb pass. These passes became the most hated and despicable symbols of apartheid. Violating Pass Laws Africans often violated the pass laws in order to find work and support their families and thus lived under constant threat of fines, harassment, and arrests. Protest against the suffocating laws drove the anti-apartheid struggle- including the Defiance Campaign in the early 50s and the huge womens protest in Pretoria in 1956. In 1960, Africans burned their passes at the police station in Sharpeville and 69 protesters were killed. During the 70s and 80s, many Africans who violated pass laws lost their citizenship and were deported to impoverished rural homelands. By the time the pass laws were repealed in 1986, 17 million people had been arrested.