Statistics Review

1
Statistics Review

• Random variable (r.v.)
• quantitative or
• categorical (qualitative)
• through the distribution of a random variable we
  understand it values of the r.v. and the
  probabilities it takes on these values
• another breakdown of r.v.s is into discrete and
  continuous.
• Examples sex, number of previous heart attacks,
  weight, time since heart transplant,

2
Distributions

• The probability distribution of a discrete r.v.
  is a specification of all the values it takes on
  along with the probabilities it takes on these
  values.
• examples Bernoulli r.v. coin toss
• indicator r.v. or indicator function of an event
  A, defined as
• what are the values of I? what are the
  corresponding probabilities?

3

• how do we represent the distribution of a
  discrete r.v.?
• table, list, formula, graph
• what properties does the distribution of a
  discrete r.v. have?
• all probabilities are between 0 and 1
• all probabilities sum to 1
• let the dist. of Y be as follows
• Y takes on values y1, y2, ,yn with
  probabilities p1, p2, , pn then we have 0pi1
  and

4

• A continuous r.v. X is described by its
  probability density function, f(x), which has the
  property that f(x)0 for all x and the total area
  bounded by its curve and the horizontal (x) axis
  is 1. Additionally, P(aXb) probability X is
  between a and b the area under the density
  curve between a and b. (Sketch!)
• Example is the normal r.v. whose density is
  represented by the familiar bell-shaped curve.

5

• often we evaluate probabilities for continuous
  r.v.s via their cumulative distribution function
  (cdf) F(x)
• so P(aXb)F(b)-F(a) (Sketch!)
• Now define a survival r.v. Y as a continuous r.v.
  taking its values in the interval from 0 to inf
  i.e., its values are thought of as the lifetime
  or survival time the time til death (or time
  til failure if were considering an inanimate
  object). So Y is a positive-valued r.v. with pdf
  f(y) and cdf F(y) and F(y)P(Yy)

6

• See example 1.1 for lifetime data on failure time
  in days of the carbon lining of a cell
• 1540 1415 660 999 1193 1006 869 1035 797
• 296 775 1424 1169 1500 728 670 841
• Or example 1.2 where lifetime is time from
  injection of a carcinogen to time of death in
  mice - note there is also an explanatory variable
  (categorical) that splits the mice into two
  groups, conventional and germ-free.

7

• Now define the survival (or reliability) function
  S(y) as S(y) 1- F(y) P(Ygty). In terms of the
  pdf, f, we have
• Note the following important properties of the
  survival function
• S(0) 1
• S(inf) 0
• S(b) gt S(a) for 0ltblta
• So the survival function is a monotone decreasing
  function on the interval from 0 to infinity (see
  Fig 1.1 p. 4)

8

• All the r.v.s we study will have a mean and
  variance
• for discrete r.v.s X,
• for continuous r.v.s X,
• X Bernoulli E(X)p where pP(X1) what is
  V(X)?
• X Binomial E(X)np, where n of "trials", and
  pprob. "S" on any one "trial" what is V(X)?

9

• HW Hand in on Wednesday, August 30 at the
  beginning of class
• Ignore as much as possible the difficulties
  regarding censoring of this data (discussed in
  Example 1.2, p.2) and do an analysis, using
  whatever methods you know at this timeuse
  graphically and numerically and statistically
  appropriate methods to compare the response
  variable (time to tumor onset) across the two
  levels of the explanatory variable
  (conventional and germ-free).