Sample Size, Methods of Probability Calculations and Central Limit Theorem

Running Head: PROBABILITY DISTRIBUTION
PROBABILITY DISTRIBUTION 3

Sample size, methods of probability calculations and central limit theorem

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Probability distribution refers to the presentation of all the possible results in a given experiment, inclusive of their associated chances of occurrence. It refers to a measure of the random phenomenon’s likelihood to occur. An experiment refers to a process whose results are not certain when it is repeated For example an experiment of tossing a die the sample space for the performed experiment will be 1, 2, 3, 5, and 6. Probability distributions are categorized into two discrete and continuous probability distributions. This paper discusses the sample space, approaches of calculating probabilities of an event occurring, both discrete and continuous probability distribution and central limit theorem.
According to probability theory sample space of a given experiment refers to a set of total possible results obtained from the experiment performed (Rubinstein et al, 2011). This sample space is denoted by use of the set notation and all possible results are enumerated as elements in a given set. It is usual referring to the sample space using the labels such as S, ? and U (incase of ‘universal set’).
For instance, if an experiment to be performed is to toss a coin, sample space will be typically the head and tail. If two coins are tossed, the sample space which will correspond will be: head and head, head and tail, tail and head, tail and tail. If a six sided die is tossed the sample space will be 1, 2, 3,4,5,6 of which the outcomes of interest will be numbers of the pips which will be facing up (Weiss et al, 2012).
A proper-defined sample space is one of the three common elements in the probabilistic model the two other elements include properly-defined set of the possible happenings and the probability given to each happenings. In many experiments, more than one sample space may be there, depending on which outcome is of more interest to the person performing the experiment. For instance, when a person is drawing from a complete deck having fifty-two cards one card, a possibility which can be on the sample space could be ranks like Ace through king, others could be various suites including: the hearts, the diamonds, the clubs and the spades. More complete outcomes that are described well could show that both the suit and denomination and the sample space explaining every card could be formed as a Cartesian item made of two sample spaces shown above (Weiss et al, 2012).
In probability theory, there are different methods used to calculate the likely of an event happening and these methods include: the frequency relative, the classical method and the subjective probabilities method.
The classical approach for calculating probabilities of items requires outcomes which are likely equal. An experiment having outcomes which are likely equal is when every simple event is having the same chances of happening (Otto, 2008). To compute chances of an event occurring using classical approach we use the following formula:
If a given experiment is carried on and it has n simple events which are likely equal and if the quantity of ways the event E can happen is m, then the chances of E occurring is computed as follows (Weiss et al, 2012):
Probability of E = Quantity of ways an event E can occur/Quantity of possible results
=m/n
Thus, if S is the space of the sample in the experiment carried on, hence
Probability of E = N (E)/N(S)
In using subjective probability the chances of an event occurring depends on an experimenter’s judgments and guesses. For instance an experimenter says that there is 40% probability of sun tomorrow. This is his guesses and he is not sure of the next day’s happenings it might rain or it might be sunny.
Using relative frequencies method in calculating the events’ occurring probability is whereby the probability of an event occurring is done basing on experience and can be verified by using scientific experimentation and this can be done as follows:
This is where the chance of a happening E is the estimated number of frequencies happening E is seen divide it by quantity of the experiment’s repetitions (Rubinstein et al,2011).
Probability of event E= Event E’s relative frequencies
=frequencies of event E/Quantity of experiment’s random variables
Discrete probability distribution refers to distributions which model discrete random variables. A random variable that is discrete is the one whose values assume whole numbers only and they are always gotten as the measure of counting, for instance: numbers of cars, buildings, persons. Here the events are independent and are always mutually exclusive, the total of probability of x for x events has a sample space of one (Rubinstein et al, 2011). Types of discrete probability distributions include: poisons probability distribution, binomial probability distribution and Geometric probability distributions.
Continuous probability distribution models the continuous random variables (Rubinstein et al, 2011). A continuous random variable has values which are continuous in nature in that it’s values are non-whole numbers such as 0.4, 0.8. They are obtained as a result of measurements for example temperature, weight and length. Types of continuous probability distributions include: Exponential probability distribution, the normal probability distribution and uniform continuous probability distributions (Otto, 2008).
Normal probability distribution is a distribution which models the chances of a wide range of variables ranging from discrete to continuous. The distribution is completely defined when the mean and the standard deviation are specified.
The central limit theorem states that if ? represents the mean of the sample having n as the size of the sample which is taken from the population having mean µ and a known variances which is ?2, then Z? which is equal to (?-µ)/(?/?n) will then follow normal distribution which is standard when n tends to infinity (Rubinstein et al, 2011). This central limit theorem can be used in sampling a population whose distribution is unknown or whom distribution is not normal (Otto, 2008).
For example to calculate the probability that the mean sample is greater than 105 trips, given that the mean is 100 trips made and the variance is 121 trips. Sample used is 36 Lorries which is taken from a population which is non-normal. Solution will be;
P (??105) = P (?-µ/ (?/?n) ?105-100/ (11/?36)
= P (Z??2.73)
= 1-(Z??2.73)
= 1-0.9968
=0.0032
Hence the probability of the mean of the sample being more than 105 trips is 0.0032

References
Weiss, N. A., & Weiss, C. A. (2012). Introductory statistics. London: Pearson Education
Rubinstein, R. Y., & Kroese, D. P. (2011). Simulation and the Monte Carlo method (Vol. 707). John Wiley & Sons.
Otto, K. (2008). Volumetric modulated arc therapy: IMRT in a single gantry arc. Medical physics, 35(1), 310-317.