Assignmenttopic: ProbabilityCoursetitle: StatisticsCoursecode: SGS116Lecturegroup: CSubmittedto: DR. Ahmed RefaatPreparedby: Mohamed Hassan Kamal /180449Assignment TopicProbabilityWhatis meant by probability?Mathematicallyit could be expressed as the possibility of occurrence of an event divided bythe total number of options you have.Or itcould be simply: the possibility of occurrence of something.Theoremsof probability:Thereare several theorems such as addition, multiplication, and Bayes’ theorem.
Thesetheorems are found because the probability definition cannot be used to findthe the probability of occurrence of at least one of the given events.Firstlythe addition theorem:It has many cases such as: 1.Mutually exclusive2.
Mutually exhaustiveMutuallyexclusive events:-Atwo events are said to be mutually exclusive if they do not have any commonelement that is to say if the possibility of event prevents the happening ofthe other.e.g.:the event of appearance of 2 heads or two tails after tossing a coinMutuallyexhaustive events:-Atwo events are called mutually exhaustive if the possibility of occurrence ofone of these events is certain (i.e.: P (XUY) =1)e.
g.:the event of having head or having tail on tossing a coin.Secondly:The multiplication theorem:If Xand Y are two events in the same sample space where P(X) ?0 and P(Y) ?0, thenP (XUY)=P(X)*P (B?A) =P (B)*P (A?B)So afterknowing multiplication theory, we have to know that there is a case derivedfrom it called independent events.Independentevents: if X and Y are not affected by the occurrence of each other then theyare called independent events.P (X?Y)=P(X)*P(Y) (where X and Y are not equal to zero)N.B:1.
if 3 events are independent then P(A?B?C)=P(A)*P(B)*P(C) 2.IF X and Y are any two events thenP(AUB)=1-P(A’)-P(B’)Third theorem: Bayes’ theorem:Thistheorem was named by the scientist Thomas Bayes who was the first one toprovide a formula that allow new evidence to bring up-to-date the beliefs.And thisformula was developed by Pierre Simon Laplace.
Thistheorem is used in the following:· Descriptionof the probability of an event based on past knowledge related to theconditions that might have relation with the event.· It isused in drug testing.The formula (simplest one):P (A?B) = (P (A?B)*P (A))/P (B) (A and B are two events ?0)The types of random variables:There are mainly two types:1. Discrete variable.2. Continuous variable.The discrete variable:It is a type of random variable that has either a certain number ofpossible values or infinite sequence of countable real numbers.
From its types: Poissondistribution and binomial distribution. Value of X X1 X2 X3 Xi Probability P1 P2 P3 Pi But this type of variables require two things:1. Every probability must lie between 0&1.2.
?P=1 The continuous variable:It is a variable that takes all values in an interval of numbers.itis characterized by Being uncountable, described by density curve.From its types: normaldistribution, uniform distribution, and exponential distribution.
Densitycurve (fig1)Probability distribution types:There lots of distributions in probabilitythe most common are Poisson, binomial, normal, exponential distribution1. Binomial distribution: it tests the possibility of eventhappening many times exceeding the number of trials and the given probabilityin each one.2. Poisson distribution: it shows the probability of a givennumber of elements in a fixed interval.3. Normal distribution: It is the most common useddistribution as it is used in many vital fields such as science and finance.itis characterized by having mean and standard deviation.
4. Exponential distribution: it is the distribution that isrelated to Poisson distribution as it expresses the time between intervals in Poissonpoint process.