(a) In probability theory, what is the meaning of the term \"event.\" [A one sen
ID: 3227220 • Letter: #
Question
(a) In probability theory, what is the meaning of the term "event." [A one sentence answer ... nothing longer!]. (b) State the definition of the conditional probability of an event A conditioned on the event B. (c) State the density function of a Gaussian random variable with mean 1 and variance 2. Express its probability distribution function in terms of Q(x). (d) State the definition of the covariance of two (possibly complex-valued) random variables X and Y. (e) What are the two properties that the joint density function of three random variable f_XYZ(x, y, z) must possess? (f) How can the joint density function of the two random variable X and Y be found from the joint density function of three random variables f_XYX(x, y, z)? (g) A chi^2-goodness of fit test is conducted for an exponential distribution with mean 10 using a histogram of 10 bins. The desired level of significance for the test was 2.5%. It was found that the test statistic had a value of 18.24. State carefully what conclusion can be drawn from this result?Explanation / Answer
A ANS) GENERAL DEFINITION IS an event is a set of outcomes of an experiment (a subset of the sample space) to which a probabilityis assigned.
B ANS) The conditional probability of an event B is the probability that the event will occur given the knowledge that an event A has already occurred. This probability is written P(B|A)
C ANS) A standard normal or Gaussian random variable is one with density (x) := 1 2 e 1 2 x2 . Its distribution function is (x) = R x (t)dt and its tail distribution function is denoted (x) := 1(x). If Xi are i.i.d. standard normals, then X = (X1,...,Xn) is called a standard normal vector in Rn. It has density n i=1 (xi) = (2)n/2 exp{|x| 2/2} and the distribution is denoted by n, so that for every Borel set A in Rn we have n(A) = (2)n/2 R A exp{|x| 2/2}dx.
D ANS)
Definition. Let X and Y be random variables (discrete or continuous!) with means X and Y. The covariance of X and Y, denoted Cov(X,Y) or XY, is defined as:
Cov(X,Y)=XY=E[(XX)(YY)]
That is, if X and Y are discrete random variables with joint support S, then the covariance of X and Y is:
Cov(X,Y)=(x,y)S(xX)(yY)f(x,y)
And, if X and Y are continuous random variables with supports S1 and S2, respectively, then the covariance of X and Y is:
Cov(X,Y)=S2S1(xX)(yY)f(x,y)dxd
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.