Introduction to Financial Mathematics
Table of Contents
1. Finite Probability Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2. Elements of Continuous Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3. Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Lecture Notes — MAP 5601 map5601LecNotes.tex i 8/27/2003
1. Finite Probability Spaces
The toss of a coin or the roll of a die results in a finite number of possible outcomes.
We represent these outcomes by a set of outcomes called a sample space. For a coin we
might denote this sample space by {H, T} and for the die {1, 2, 3, 4, 5, 6}. More generally
any convenient symbols may be used to represent outcomes. Along with the sample space
we also specify a probability function, or measure, of the likelihood of each outcome. If
the coin is a fair coin, then heads and tails are equally likely. If we denote the probability
measure by P, then we write P(H) = P(T) = 1
2 . Similarly, if each face of the die is equally
likely we may write P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = 1
6 .
Defninition 1.1. A finite probability space is a pair (
, P) where
is the sample space set
and P is a probability measure:
If
= {!1, !2, . . . , !n}, then
(i) 0 < P(!i) 1 for all i = 1, . . . , n
(ii)
n Pi=1
P(!i) = 1.
In general, given a set of A, we denote the power set of A by P(A). By definition this
is the set of all subsets of A. For example, if A = {1, 2}, then P(A) = {;, {1}, {2}, {1, 2}}.
A researcher determines that 42.7% of all downtown office buildings have ventilation problems. Is this a statistic or a parameter; explain your answer.
The Healthcare Professional that I chose is a Dental Assistant. The reason why I chose to use a dental assistant as the Healthcare Professional was because I think that they are very important along with the dentists. Without dental assistants, dentist’s jobs would be more stressful than it already is in their job. Dental assistants greatly increase the efficiency of the dentist in the delivery of quality oral health care are valuable members of the dental care team.
Ott, L., & Longnecker, M. (2010). A Continuous Probability Distribution. An introduction to statistical methods and data analysis (6th ed., p. 175). Belmont, CA: Brooks/Cole Cengage Learning.
Chance. 50/50. 1:2. Odds. These terms are familiar in gambling. Bet it all give it a shot. Is it worth the consequences? Are the problems worth the rewards? Imagine a gamble between life and death, war and peace. Would it be worth the destruction to have your way? What would you do to keep a competitor out of the game? Going neck and neck to find a way around combat. Would the world be the same? What would happen if you lost? When tension between World War II grows, a gamble for nuclear arms rises, becoming the cold war.
event such as a sequence of numbers as produced by a random event generator. The
Introduction to the basic concepts of probability and statistics with discussion of applications to computer science.
Inferential statistics establish the methods for the analyses used for conclusions drawing conclusions beyond the immediate data alone concerning an experiment or study for a population built on general conditions or data collected from a sample (Jackson, 2012; Trochim & Donnelly, 2008). With inferential statistics, you are trying to reach conclusions that extend beyond the immediate data alone. For instance, we use inferential statistics to try to infer from the sample data what the population might think. A requisite for developing inferential statistics supports general linear models for sampling distribution of the outcome statistic; researchers use the related inferential statistics to determine confidence (Hopkins, Marshall, Batterham, & Hanin, 2009).
Social Workers main objectives are to work for Social Justice, and invoke positive changes for future generations. Social work as I have come to know it is a helping profession on many levels. Social work is not concentrated on only one specific area of need, but targets many areas on a larger scale with vast opportunities in the field. As Hick (2006) states
To prove the two hypothesis the researchers present the participants with a series of opportunities to cheat, but then surprise them with an additional series of cheating opportunities. To increase their interest in participating the contributors obtain $.10 for each flip of a coin, but only if the side up of the coin is “heads”. The participants have the opportunity of flip the coin 7, 10 or 13 times, the flipping of the coin happens privately, and they are told not to cheat. The researches then tried to prove the hypothesis in 4
Perhaps one of the most versatile and diverse careers is social work, as this career works with individuals struggling with substance abuse, low-income families, and homeless individuals. Social work is a profession that is concerned with solving personal, group, and community relationships (Farley, Smith, & Boyle, 2011). Among some of the most important aspects of social work are the restoration of impaired social functioning, the provision of social services, and prevention (Farley et al., 2011). Reducing problems in human relationships and improving human interactions among individuals are the major focuses of this profession, regardless of which people group the social workers are serving (Farley et al., 2011). Even though it is a relatively new profession, social work has existed in various forms throughout history. However, the beginning of social work’s development as a profession was primarily European in origin, with the ideals spreading to the United States
Throughout the Social Work Program at Winona State University-Rochester, I have been able to expand my knowledge about what it means to be a social worker. Prior to joining this program, when I thought of a social worker, I thought of someone who is highly motivated, compassionate, supportive, and someone who wants to make a difference in somebody’s life. Although this definition is true, overtime I have been able to expand this definition. Now, they are also someone who builds relationships with clients during what are often their lowest points in their lives and they are a person who often will put others needs before their own. Social workers use their knowledge about maintaining a person’s dignity and worth and rely on skills that help them understand people within their environment, and using their training to help people from all walks of life both on the micro level and the macro level.
Probability is always surrounding us from stock markets to the ever-simple heads or tails. This very complicated area of mathematics can be explained in a simpler way. It is how likely an event is to happen. The probability of an event will always be between 0 and 1. The closer it is to one, the more likely the event is to happen.
I would to begin this paper with an authentic definition of the Social Work field. Social work practices involves facilitating change—in other words, working with others, not doing something to them or for them(Dubois). Most people and social workers would always use the world ‘help’ in some way or form when defining social workers. Which is true but the confusion come in when asked where they work, who they assist and how they differ from other helping professions. Social workers can be sustain abuse therapists, child welfare specialists, and school social workers. We are not limited to just the department of social services. They can work in nursing homes, hospitals, and even in legislation.
Medical social work is a sub-discipline of social work. The medical social worker is a member of an interdisciplinary team that collectively participate in patient care and advocacy. In addition to the social worker, a medical interdisciplinary team can consist of physicians, nurses, physical,
Social Workers also approach the clients to not only resolve personal conflicts but to allow the client to function best within their environments. Social Work distinguishing characteristics include the emphasis on the person-in-environment model as well as its emphasis on social justice. Compared to other similar professions, Social Work has differing professional values and ethics that include appreciation to human diversity, understanding of social welfare policy and services, emphasis on both social and economic justice, as well as a strong foundation in education. The importance of Social Work includes numerous of explanations. Social Workers help countless of people navigate through major life challenges or setbacks to find hope as well as options to achieve greater things. Not only do Social Workers help individuals overcome problems but they also stand up for human rights and social justice to assist