Chapter 7_Random Variables - Dr by hcj


									                             Introduction and
7.1 Discrete and Continuous Random Variables
Random Variable
 A Random Variable is a variable whose value is a
  numerical outcome of a random phenomenon.
 In this section, we will learn two ways of assigning
  probabilities to events. These two models will
  dominate our application to probability to statistical
Discrete Random Variables
 A Discrete Random Variables X has a countable
 number of possible values.
 The Probability distribution of X lists the values and
 their probabilities.

       Value     X1     X2    X3   …      Xk
       of X
       Probabi   P1     P2    P3   …      Pk
Example 7.1 Getting Good Grades
 An instructor of a large class assigns grades in the
  following manner. What is the Probability that a
  student gets a B or better?
Grade 0            1         2         3         4
      (F)          (D)       (C)       (B)       (A)
Probab    0.10     0.15      0.30      0.30      0.15
 P(grade is 3 or 4) =
 P(X = 3) + P(X=4)=
 0.3+0.15=
 0.45
Probability Histograms
                                 Random Digits
         1        2          3        4       5       6       7       8   9

                       Random Digits

                                 Benford’s Law


             0     1         2    3       4       5       6       7   8   9
             Benford’s law
Example 7.2 Tossing Coins
 What is the probability distribution of the discrete
 random variable X that counts the number of heads in
 four tosses of a coin?
                HTTT    THTH     HHHT
                THTT    HHTT     HHTH
                TTHT    THHT     HTHH
       TTTT     TTTH    TTHH     THHH     HHHH
       X=0      X=1     X=2      X=3      X=4
Probability of each value of X
 P(X=0)=1/16=0.0625          P(X=1)= 4/16=0.25
 P(X=2)= 6/16=0.37           P(x=3)=4/16= 0.25
 P(X=4) =1/16=0.0625
# of heads
              0       1      2       3        4
Probability   0.625   0.25   0.37    0.25     0.625
Continuous Random Variables
 Choosing a random # between 0 and 1.
 S={all numbers x such that 0≤x≤1}
 We cannot assign probabilities to each individual value
  of x and then sum, because there are infinitely many
  possible values.
 Instead we use areas under a density curve to assign
  probability to events.
 Any density curve has an area of exactly 1 underneath
  corresponding to a probability of one.
Example 7.3 Random Numbers and
the Uniform Distribution
P(0.3≤X≤0.7)   1   P(X≤0.5 or X>0.8)
                   =0.5 +0.2= 0.7
The probability distribution of a continuous random
variable assigns probabilities as area under a density
Continuous Random Variable
 A Continuous Random Variable X takes all values
  in an interval of numbers. The Probability
  distribution of X is described by a density curve.
 The probability of any event is the area under the
  density curve and above the values of X that make up
  the event.
 All continuous probability distributions assign
  probability 0 to every individual outcome.
 We can ignore the distinction between > and ≥ when
  finding probabilities for continuous(but not discreet)
  random variables.
Normal Distributions as Probability
 Because any density curve describes an assignment of
  probabilities, normal distributions are probability
 Recall that N(µ,σ) stands for a normal distribution
  with mean µ and standard deviation σ.
 If X has the N(µ,σ) distribution, then the standardized
  variable          Z=X-µ
is a standardized normal random variable having the
  distribution N(0,1).
Example 7.4 Drugs in Schools
 N(0.3,0.0118)
 What is the probability that the poll results differ from
  the truth about the population by more than 2
  percentage points.
 P. 401-404
 7.6, 7.10, 7.13, 7.15

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