Out of Shadows Essays

Out of Shadows Essays Out of Shadows Essay Out of Shadows Essay Change and racism evidently aids the characterisation and the development of the text because these significant themes help the portrayal of the characters as they are relatable in society. The text being explored is out of the Shadows by Jason Wallace. Change reoccurs throughout the text which helps to develop Roberts character. Similarly, the major connotation of racism also engages readers to relate to the changes of Roberts and Ivans character. Change is evident in the development of Roberts character as he experiences his journey with Ivan and his friends.At the beginning of the text, Out of the Shadows by Jason Wallace, Robert has an awkward confrontation with his ex-friend Nelson when he was with Ivan. Knowing how Ivan would react, Robert harshly told Nelson to leave him alone and not to mind his business. This is demonstrated in the quote, I dont know why I felt surprised, wasnt that what Id been after? The use of the soliloquy creates an impression t hat Robert is realising he is changing into to someone he isnt. He is questioning why he was having doubts as that was what he had wanted to achieve in the first place.This obscurity in how Robert feels about becoming a new self helps the idea that change is a recurring theme throughout the text. In another scenario, Robert is responding to his dad why he has not been playing or hanging around with Nelson anymore. In this quote, Hes not really a friend anymore. Robert replied with a tone that seemed he was trying to tell himself how he felt instead of his father. The doubt he is experiencing emphasises he is trying to turn over a new leaf in order to fit in, and hence Robert deals with change throughout the text. Another significant theme recurring through the text is racism.This is shown in the quote, Hes using his colour to his advantage. The metaphor used suggests that the author is putting out the fact that racism was still a major issue in those times and needed to be viewed d ifferently. Ivan is disgusted by Nelson, who he says is using is colour to his advantage because blacks in those time were viewed as the unfortunate. The harsh meaning behind this racist comment, gives people an urge to change and make a difference in racial equality. Ivans character persists to tell the readers he does not appreciate the blacks and always finding a reason to hate on them. This is shown in the line No, special.Hes black. The italics used for the word, special emphasises racial equality is not evident in this text. The composers use of a metaphor shows the scorn and harsh personality of Ivan and thus engages the audience in the relevant issues of society. In conclusion, change and racism are significant themes in the text which are sadly still relevant now in our society. The change in Robert’s character gives the reader’s a personal insight on how these relevant ideas involving post-colonial Zimbabwe, affect people negatively. It is responsible for t he portrayal of all the characters in ‘Out of Shadows’.

Expected Value of a Binomial Distribution

Expected Value of a Binomial Distribution Binomial distributions are an important class of discrete probability distributions. These types of distributions are a series of n independent Bernoulli trials, each of which has a constant probability p of success. As with any probability distribution we would like to know what its mean or center is. For this we are really asking, â€Å"What is the expected value of the binomial distribution?† Intuition vs. Proof If we carefully think about a binomial distribution, it is not difficult to determine that the expected value of this type of probability distribution is np. For a few quick examples of this, consider the following: If we toss 100 coins, and X is the number of heads, the expected value of X is 50 (1/2)100.If we are taking a multiple choice test with 20 questions and each question has four choices (only one of which is correct), then guessing randomly would mean that we would only expect to get (1/4)20 5 questions correct. In both of these examples we see that  E[ X ] n p. Two cases is hardly enough to reach a conclusion. Although intuition is a good tool to guide us, it is not enough to form a mathematical argument and to prove that something is true. How do we prove definitively that the expected value of this distribution is indeed np? From the definition of expected value and the probability mass function for the binomial distribution of n trials of probability of success p, we can demonstrate that our intuition matches with the fruits of mathematical rigor. We need to be somewhat careful in our work and nimble in our manipulations of the binomial coefficient that is given by the formula for combinations. We begin by using the formula: E[ X ] ÃŽ £ x0n x C(n, x)px(1-p)n – x. Since each term of the summation is multiplied by x, the value of the term corresponding to x 0 will be 0, and so we can actually write: E[ X ] ÃŽ £ x 1n x C(n , x) p x (1 – p) n – x . By manipulating the factorials involved in the expression for C(n, x) we can rewrite x C(n, x) n C(n – 1, x – 1). This is true because: x C(n, x) x n!/(x!(n – x)!) n!/((x – 1)!(n – x)!) n(n – 1)!/((x – 1)!((n – 1) – (x – 1))!) n C(n – 1, x – 1). It follows that: E[ X ] ÃŽ £ x 1n n C(n – 1, x – 1) p x (1 – p) n – x . We factor out the n and one p from the above expression: E[ X ] np ÃŽ £ x 1n C(n – 1, x – 1) p x – 1 (1 – p) (n – 1) - (x – 1) . A change of variables r x – 1 gives us: E[ X ] np ÃŽ £ r 0n – 1 C(n – 1, r) p r (1 – p) (n – 1) - r . By the binomial formula, (x y)k ÃŽ £ r 0 kC( k, r)xr yk – r the summation above can be rewritten: E[ X ] (np) (p (1 – p))n – 1 np. The above argument has taken us a long way. From beginning only with the definition of expected value and probability mass function for a binomial distribution, we have proved that what our intuition told us. The expected value of the binomial distribution B( n, p) is n p.