Chap 4. 확률과 확률분포 Probability


4.1 서론

4.2 확률의 정의
  • Def. Sample space S is the set of all possible outcomes of that experiment
  • Def. Event is any collection(subset) of outcomes contained in the sample space
  • Def. P(A) is called the probability of the event.
      1. For any event A, P(A)>=0
      2. P(S)=1
      3. If A_1,A_2,A_3,... is an infinite collection of mutually exclusive events, then P(A_1 \cup A_2 \cup A_3 \xup cdots) = \sum_{i=1}^\inf P(A_i)


4.3 확률의 계산
  • Def. Any ordered sequence of k objects taken from a set of n distinct object is called a permutation of size k of the objects.
  • Def. Given a set of n distinct objects, any unordered subset of size k of the objects is called a combination

4.4 확률의 법칙
  • Prop. For any event A and B, P(A \cup B)=P(A)+P(B)-P(A \cap B)
  • Prop. If A and B are mutually exclusive, then P(A \cap B)=0, P(A \cup B)=P(A)+P(B)
  • Def. For any two events A and B with P(B)>0, the conditional probability of A given that B has occured is defined by P(A|B)=P(A \cap B) / P(B)
  • The multiplication Rule : P(A \cap B) = P(A|B)P(B)
  • The law of total probability : Let A_1, A_2, ... A_n be mutually exclusive and exhaustive events. Then for any other event B, P(B) = P(B|A_1 )P(A_1 ) + \cdots + P(B|A_n )P(A_n) = \sum_{i=1}^n P(B|A_i)P(A_i)
  • Def. Two events A and B are independent if P(A|B)=P(A) and are dependent otherwise.
  • Prop. A and B are independent iff P(A \cap B) = P(A)P(B)
  • Def. Events A_1, A_2, ... A_n are mutually independent if for every and every subset of indices i_1, i_2, ... i_k P(A_i_1 \cap A_i_2 ... \cap A_i_k) = P(A_i_1)P(A_i_2)...P(A_i_k)
  • Prop. For any event A, P(A)=1-P(A`)

4.5 확률의 법칙
    Thm. Bayes' Theorem :
    Let A_1,A_2, ...A_n be a collection of n mutually exclusive and exhaustive events with P(A_i)>0 for i=1,2,...n. Then for any other event B for which P(B)>0

4.6 이산확률변수
  • Def. For a given sample space S of some experiment, a random variable is any rule that associates a number with each outcome in S.
  • Def. Any random variable whose only possible values are 0 and 1 is called a Bernoulli random variable
  • Def. A set is discrete either if it consists of a finite number of elements, or if its can be listed so that there is a first element, a second element, a third element, and so on, in the list.
  • Def. A random variable is said to be discrete if its set of possible values is a discrete set.
  • Def. The probability distribution or probability mass function(p.m.f.) of a discrete random variable is defined for every number x by P(x)=P(X=x)=P(all s \in S : X(s)=x)
  • Def. The cumulative distribution function(c.d.f.) F(x) of a discrete random variable X with p.m.f. P(x) is defined for every number x by F(x)=P(X<=x)= \sum_{y<=x}P(y)
  • Prop. For any two numbers a and b with a<=b, P(a<=X<=b)=F(b)-F(a-)
  • Def. The expected value or mean value of X, denoted by E(X) or \mu_x ,is E(X)=\mu_x = \sum x*P(x)
  • Prop. The expected value of a function h(X), denoted by E[h(X)] or \mu_h(x), is E[h(X)]=\mu_h(x)=\sum h(x)*P(x)
  • Prop. E(aX+b)=aE(X)+b
  • Def. The variance of X, denoted by V(X) or \sigma_x^2, is V(X)=\sum(x-\mu)^2*P(x)=E[(X-\mu)^2]
  • Prop. V(X)=E(X^2)-[E(X)]^2
  • Prop. V(aX+b)=a^2V(X)

4.7 두 확률변수의 결합분포
  • Def. Let X and Y be two discrete random variables defined on the sample space of an experiment. The joint probability mass function P(x,y) is defined for each pair of numbers (x,y) by P(x,y)=P(X=x and Y=y).
    Let A be any set consisting of pairs of (x,y) values. The the probability P[(X,Y) \in A] = \sum\sum_{(x,y) \in A} P(x,y)
  • Def. The marginal probability mass function of X and of Y, denoted by P_x(x) and P_y(y), respectively, are given by P_x(x)=\sum_y P(x,y), P_y(y)=\sum_xP(x,y)
  • Prop. E[h(X,Y)]=\sum_x \sum_y h(x,y)*P(x,y) if X and Y are discrete
  • Def. The covariance between two random variables X and Y is Cov(X,Y)=E[(X-\mu_x)(Y-\mu_y)]
  • Prop. Cov(X,Y)=E(XY)-\mu_x*\mu_y
  • Def. The correlation coefficient of X and Y, denoted by Corr(X,Y) or \rho is defined by \rho = Cov(X,Y) \over {\sigma_x \sigma_y}
  • Prop.
      1. Corr(aX+b, cY+d)=Corr(X,Y)
      2. -1<=Corr(X,Y)<=1
  • Prop.
      1. If X and Y are independent, then \rho=0, but \rho=0 does not imply independence
      2. \rho=1 or -1 iff Y=aX+b for some numbers a and b with a \ne 0

Quiz 2. Email: lbg@kowon.dongseo.ac.kr
    1. If the sample space is S=C_1 \cup C_2 and if P(C_1)=0.8 and P(C_2)=0.5, find P(C_1 \cap C_2).

    2. Let the probability set function P(A) of two random variable X and Y be P(A)=\sum\sum_A f(x,y) where f(x,y)=1/52, {(x,y); (x,y) = (0,1),(0,2),...,(0,13),(1,1),...,(1,13),...,(3,13)}.
    Compute P(A) when A = {(x,y); x+y=4, (x,y) \in S }

    3. Let a fair coin be tossed at random on successive independent trials. Find the probability that the first head appears on the third trials.

    4. A study of the relationship among adult drivers between income level(L=low, M=medium, and H=high) and preference for one of the "big three" automobile manufactures(denoted by A, B, and C here) yield the accompanying table of joint probabilities.
    LMHSum
    A0.100.130.020.25
    B0.200.120.080.40
    C0.100.150.100.35
    Sum0.400.400.20

    Use this table to compute the following conditional probabilities.
      a.P(B|H) b.P(M|C) c.P(A'|M)
      d.P(M|A') e.P(M|B∪C) f.P(L∪M|C)

    5. An executive has both a morning and an afternoon meeting on a particular day. Let A = {late to the morning meeting} and B={late to the afternoon meeting}.
      a. If P(A)=0.4, P(B)=0.5, and P(A∩B)=0.25, are A and B independent events?
      b. If A and B are independent events with P(A)=0.4 and P(B)=0.5, what is the probability that the executive is on time to both meetings?

    6. A restaurant serves three fixed-price dinners costing $7, $9, and $10. For a randomly selected couple dining at this restaurant, let X = the cost of the man's dinner and Y = the cost of the woman's dinner. Suppose the joint p.m.f. of X and Y is given in the following table
    P(x,y)7910
    70.050.050.10
    90.050.100.35
    100.000.200.10
      a. Compute the marginal p.m.f.'s of X and Y
      b. What is the probability that the man's and the woman's dinner cost at most $9 each?
      c. Compute the correlation coefficient of X and Y. d. Are X and Y independent?
      e. What is the expected total cost of the dinner for the two people?

    7. Incidence of rare disease. Only one in 1000 adults is afflicted with a rare disease for which a diagnostic test has been developed. The test is such that, when an individual actually has the disease, a positive result will occur 99% of the time, while an individual without the disease will show a positive test result only 2% of the time. If a randomly selected individual is tested and the result is positive, what is the probability that the individual has the disease?

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