同济大学：《Matlab在机械设计中的应用》课程电子教案（PPT课件）Chapter 09 Probability and statistics

Probability and Statistics(1/35) Concept Probability is a measure of the likelihood that some event will occur.It is also a way to quantify or to gauge the likelihood that an observed measurement or random variable will take on values within some set or range of values. A random experiment is defined as a process or action whose outcome cannot be predicted with certainty and would likely change when the experiment is repeated. >The sample space is the set of all outcomes from an experiment. It is possible sometimes to list all outcomes in the sample space. @月两大学 AW TONGJI UNIVERSITY

➢ Probability is a measure of the likelihood that some event will occur. It is also a way to quantify or to gauge the likelihood that an observed measurement or random variable will take on values within some set or range of values. ➢ A random experiment is defined as a process or action whose outcome cannot be predicted with certainty and would likely change when the experiment is repeated. ➢ The sample space is the set of all outcomes from an experiment. It is possible sometimes to list all outcomes in the sample space. Probability and Statistics(1/35) Concept

Probability and Statistics(2/35) Concept To find the probability that a continuous random variable falls in a particular interval of real numbers,we have to calculate the appropriate area under the curve of f). This is represented by: b P(a≤X≤b)=∫fx)dr CDHAW @月停大学 TONGJI UNIVERSITY

➢ To find the probability that a continuous random variable falls in a particular interval of real numbers, we have to calculate the appropriate area under the curve of f(x). This is represented by: Probability and Statistics(2/35) Concept

Probability and Statistics(3/35) Concept The cumulative distribution function F(x)is defined as the probability that the random variable X assumes a value less than or equal to a given x. This is calculated from the probability density function,as follows: F(x)=P(X≤x)=∫ft)d. @日济大学 TONGJI UNIVERSITY

➢ The cumulative distribution function F(x) is defined as the probability that the random variable X assumes a value less than or equal to a given x. This is calculated from the probability density function, as follows: Probability and Statistics(3/35) Concept

Probability and Statistics(4/35) Concept This shows the probability density function with the associated cumulative distribution function.Notice that the cumulative distribution function takes on values between 0 and 1. PDF CDF 0.4 0.3 0.8 0.6 80.2 0.4 0.1 0.2 0 0 -2 0 4 0 2 CDHAW @月停大学 TONGJI UNIVERSITY

➢ This shows the probability density function with the associated cumulative distribution function. Notice that the cumulative distribution function takes on values between 0 and 1. Probability and Statistics(4/35) Concept

Probability and Statistics(5/35) Concept The mean or expected value of a random variable is defined using the probability density (mass)function.It provides a measure of central tendency of the distribution. u=EX]=∫xfx)dx We see from the definition that the expected value is a sum of all possible values of the random variable where each one is weighted by the probability that x will take on that value. 日濟大学 AW TONGJI UNIVERSITY

➢ The mean or expected value of a random variable is defined using the probability density (mass) function. It provides a measure of central tendency of the distribution. We see from the definition that the expected value is a sum of all possible values of the random variable where each one is weighted by the probability that X will take on that value. Probability and Statistics(5/35) Concept

Probability and Statistics(6/35) Concepta酸 To calculate this in MATLAB,one can use the function called mean.If the argument to this function is a matrix,then it provides a vector of means,each one corresponding to the mean of a column. > One can find the mean along any dimension (dim)of multi- dimensional arrays using the syntax:mean(x,dim) IfX=[012,345]; then mean (X,1)is[1.52.53.5] CPHAW and mean (X2)is[1;4] @月停大学 TONGJI UNIVERSITY

If X = [0 1 2; 3 4 5]; then mean (X,1) is [1.5 2.5 3.5] and mean (X,2) is [1; 4] ➢ To calculate this in MATLAB, one can use the function called mean. If the argument to this function is a matrix, then it provides a vector of means, each one corresponding to the mean of a column. ➢ One can find the mean along any dimension (dim) of multi￾dimensional arrays using the syntax: mean(x, dim). Probability and Statistics(6/35) Concept

Probability and Statistics(7/35) Concept The variance of a random variable is given by the following definition. G2=V(X)E[(X-u)]=f(x-u)f(x)dx We note that equation can also be written as: V(X)=E[X2]-u2=E[X2]-(E[X])2 @日济大学 AW TONGJI UNIVERSITY

➢ The variance of a random variable is given by the following definition. We note that equation can also be written as: Probability and Statistics(7/35) Concept

Probability and Statistics(8/35) Concept These statistics can be calculated in MATLAB using the functions std(x)and var(x),where x is an array containing the sample values. Similar to the function mean,these can have matrices or multi- dimensional arrays as input arguments. CDHAW @月协大学 TONGJI UNIVERSITY

➢ These statistics can be calculated in MATLAB using the functions std(x) and var(x), where x is an array containing the sample values. ➢ Similar to the function mean, these can have matrices or multi￾dimensional arrays as input arguments. Probability and Statistics(8/35) Concept

Probability and Statistics(9/35) Axioms >Probabilities follow certain axioms that can be useful in computational statistics.We let S represent the sample space of an experiment and E represent some event that is a subset of S. The probability ofevent E must be between 0 and 1: 0≤P(E)≤1 P(S)=1 For mutually exclusive events: P(EE2...E)=>P(E i=1 @日济大学 AW TONGJI UNIVERSITY

Probability and Statistics(9/35) Axioms ➢ Probabilities follow certain axioms that can be useful in computational statistics. We let S represent the sample space of an experiment and E represent some event that is a subset of S. The probability of event E must be between 0 and 1: For mutually exclusive events:

Probability and Statistics(10/35 Axioms金a The conditional probability of event E given event F is defined as follows: EF):P(F) P(F) Here P(EF)represents the joint probability that both E and F occur together. We can rearrange this equation to get the following rule: P(E⌒F)=P(F)P(EIF) 细月济大学 TONGJI UNIVERSITY

➢ The conditional probability of event E given event F is defined as follows: Here represents the joint probability that both E and F occur together. We can rearrange this equation to get the following rule: Probability and Statistics(10/35) Axioms