Free Essays on Sociological Imagination

C. Wright Mills coined the term sociological imagination in his book The Sociological Imagination. Sociological imagination refers to the relationship between individual troubles and the large social forces that are the driving forces behind them. Mills argued that in order to avoid becoming victims of a large, seemingly distant event, we must learn to understand the relationship between private troubles and social issues. For example if a small group of people in a society were unemployed, then one would ask what it is about them that is keeping them from working. However, when millions of people in a society are unemployed, one must look at what social forces are occurring to cause such an expansive incident. An individual’s life and experiences are shaped by the society in which he or she lives. The life that I know now would be completely different if I was born about 60 years earlier. I have spent the majority of my life living in a time where the economy of the United States was on an incredible up swing. This is due in large part to the commercialization of the Internet, also known as the â€Å"Dot-Com Era.† The stock market was seeing record gains that seemed to have no end. This affected my life because it was a time when it seemed like everyone I knew, including my family, was profiting off the economy. People the ability to spend money to an extent that they would not have normally been able to. Because of this nice things surrounded everyone I knew and me. We were all getting new cars, buying new houses, buying clothes and other things of the sort. Because this society was in great shape, so were all the individuals living in it. If I had been born about 60 years earlier than my actual birth, I would have not lived through a time of economic prosperity, but rather I would have lived through The Great Depression of the 1930’s. The Great Depression was a time of economic despair. The unemployment rate was rising at a subs... Free Essays on Sociological Imagination Free Essays on Sociological Imagination C. Wright Mills coined the term sociological imagination in his book The Sociological Imagination. Sociological imagination refers to the relationship between individual troubles and the large social forces that are the driving forces behind them. Mills argued that in order to avoid becoming victims of a large, seemingly distant event, we must learn to understand the relationship between private troubles and social issues. For example if a small group of people in a society were unemployed, then one would ask what it is about them that is keeping them from working. However, when millions of people in a society are unemployed, one must look at what social forces are occurring to cause such an expansive incident. An individual’s life and experiences are shaped by the society in which he or she lives. The life that I know now would be completely different if I was born about 60 years earlier. I have spent the majority of my life living in a time where the economy of the United States was on an incredible up swing. This is due in large part to the commercialization of the Internet, also known as the â€Å"Dot-Com Era.† The stock market was seeing record gains that seemed to have no end. This affected my life because it was a time when it seemed like everyone I knew, including my family, was profiting off the economy. People the ability to spend money to an extent that they would not have normally been able to. Because of this nice things surrounded everyone I knew and me. We were all getting new cars, buying new houses, buying clothes and other things of the sort. Because this society was in great shape, so were all the individuals living in it. If I had been born about 60 years earlier than my actual birth, I would have not lived through a time of economic prosperity, but rather I would have lived through The Great Depression of the 1930’s. The Great Depression was a time of economic despair. The unemployment rate was rising at a subs...

Moment Generating Function for Binomial Distribution

Moment Generating Function for Binomial Distribution The mean and the variance of a random variable X with a binomial probability distribution can be difficult to calculate directly. Although it can be clear what needs to be done in using the definition of the expected value of X and X2, the actual execution of these steps is a tricky juggling of algebra and summations. An alternate way to determine the mean and variance of a binomial distribution is to use the moment generating function for X. Binomial Random Variable Start with the random variable X and describe the probability distribution more specifically. Perform n independent Bernoulli trials, each of which has probability of success p and probability of failure 1 - p. Thus the probability mass function is f (x) C(n , x)px(1 – p)n - x Here the term C(n , x) denotes the number of combinations of n elements taken x at a time, and x can take the values 0, 1, 2, 3, . . ., n. Moment Generating Function Use this probability mass function to obtain the moment generating function of X: M(t) ÃŽ £x 0n etxC(n,x))px(1 – p)n - x. It becomes clear that you can combine the terms with exponent of x: M(t) ÃŽ £x 0n (pet)xC(n,x))(1 – p)n - x. Furthermore, by use of the binomial formula, the above expression is simply: M(t) [(1 – p) pet]n. Calculation of the Mean In order to find the mean and variance, youll need to know both M’(0) and M’’(0). Begin by calculating your derivatives, and then evaluate each of them at t 0. You will see that the first derivative of the moment generating function is: M’(t) n(pet)[(1 – p) pet]n - 1. From this, you can calculate the mean of the probability distribution. M(0) n(pe0)[(1 – p) pe0]n - 1 np. This matches the expression that we obtained directly from the definition of the mean. Calculation of the Variance The calculation of the variance is performed in a similar manner. First, differentiate the moment generating function again, and then we evaluate this derivative at t 0. Here youll see that M’’(t) n(n - 1)(pet)2[(1 – p) pet]n - 2 n(pet)[(1 – p) pet]n - 1. To calculate the variance of this random variable you need to find M’’(t). Here you have M’’(0) n(n - 1)p2 np. The variance ÏÆ'2 of your distribution is ÏÆ'2 M’’(0) – [M’(0)]2 n(n - 1)p2 np - (np)2 np(1 - p). Although this method is somewhat involved, it is not as complicated as calculating the mean and variance directly from the probability mass function.