Probability and StatisticsModel Answer of 2024-2025 Final ExamBJTU Lancaster University CollegePart I.(28 pt)Fill in the blanks.2.Suppose that the length of the components produced by a certain machine follows a normaldistribution with parameters=10.05,and o =0.06.If it is stipulated that the length ofthe parts within a certain range (10.05+0.18)cm are qualified,then the probability that thecomponents are unqualified is 0.0026.3.Suppose X1,X2.X is a random sample from population X ~N(u,2),then4.The random variables X and Y are independent and identically distributed.Let U=X+Yand V=X-Y.Then,the correlation coefficient between U and V is 0.5.Suppose that (X1,Y1),,(Xn,Yn)is a random sample from population N(u1,u2,01,0,p).Let =(=1X-Y).Then,the variance of is26.If the random variables X1,X2...,Xn form a random sample of size n from a given distributionwith mean u and variance o2,then the central limit theorem shows that the distribution of the7.The PF of the population X is stated as follows.23Consider the following sample:1,3,2,21,21,3.The maximum likelihood estimate of isPart II.(72 pt)Problem-solving.1.(10 pt)Suppose that X and Y are two mutually independent random variables,where X~U0,1],Y~E(1).We want to find(1)the joint PDF of X and Y;(2)P{Y≤X;(3)P{X+Y≤1.ISolution:(1)The PDF of X isfx(z)=0,else.1,0