Continuous Random Variables and Their Probability Distributions

1
Chapter 4

• Continuous Random Variables and Their Probability
  Distributions

2
Sec 4.2

• Distribution Functions

3
Cumulative Distribution Function

• Definition Let Y be any random variable. The
  distribution function (cumulative distribution
  function) for Y, F(y), is defined by, F(y)
  P(Y?? y).
• Note This function is defined for both
  continuous and discrete random variables.

4
Properties of any cdf F(x)

• Let Y be a random variable with distribution
  function F(y). Then
• For all a and b, if a lt b then F(a) ? F(b).
  (F is a nondecreasing function)
• (lower
  limit of F is 0)
• (upper
  limit of F is 1)
• (F is
  right continuous)
• Let Y be be a random variable with distribution
  function F(y). Then for a lt b Pa lt Y ? b
  F(b) F(a).

5
Continuous Random Variables

• Definition A random variable Y is a continuous
  random variable if its distribution function F(y)
  is continuous for all y?R.
• Definition For a continuous random variable Y
  with distribution function F(y) the probability
  density function (pdf) of Y is defined by f(y)
  (d/dy)F(y) F'(y). Consequently,

6
Properties of pdf

• If f(y) is the density function of a continuous
  random variable Y, then
• f(y) ? 0 for all y?R
• ?f(y)dy 1 where the integral is over (-?, ?)
• P(Ya) 0 for any real number a
• If altb, then

7
Sec 4.3

• Expected Value

8
Expected Value of Y

• Definition Let Y be a continuous random
  variable with pdf f(y) then the expected value of
  Y, E(Y), is defined by E(Y) ? y f(y) dy
  provided the integral converges absolutely.
• The expected value of Y is denoted by ?.

9
Functions of Random Variables

• Theorem Let Y be a continuous random variable.
  Suppose that for some real valued function g the
  random variable X g(Y) is continuous. Then the
  expectation of X is given by E(X) Eg(Y)
  ? g(y) f (y) dy provided the integral
  converges absolutely

10
Properties

• Theorem Let c be a constant and let g1(Y),
  g2(Y),.., gk(Y) be functions of a continuous
  random variable Y. Suppose the expectations of
  g1(Y), g2(Y), , gk(Y) all exist. Then,
• E(c) c
• Ecg(Y) cEg(Y)
• E(g1(Y) g2(Y) gk(Y) ) E(g1(Y))
  E(g2(Y)) E(gk(Y))
• V(Y) E(X - ?)2 E(X2) - ?2.

11
Section 4.4-4.7

• Uniform Probability Distribution, U(?1,?2)
• Normal Probability Distribution, N(?,?2)
• Gamma Probability Distribution, Ga(?, ?)
• Chi-Square Probability Distribution, ?2(?)
• Exponential Probability Distribution, exp(?)
• Beta Probability Distribution, Beta(?, ?)
• Uniform Probability Distribution,
  U(0,1)Beta(1,1)

12
The Uniform Distribution

• The continuous random variable Y has uniform
  distribution of the interval (?1, ?2) if the
  density function of Y is f(y) 1/(?2 - ?1)
  for y ? (?1,?2) and 0 otherwise
• E(Y) (?1 ?2)/2
• V(Y) (?2- ?1)2/12

13
The Normal Distribution

• The continuous random variable Y has a normal
  distribution if the density function of Y is
  for all y? (-?,?)
• E(Y) ?
• V(Y) ?2

14
Calculation of Normal Probability Density
Function (pdf)

• Standard Normal pdf , Denoted by N(0,1)
• Z(Y-?)/?
• Find scores, given probability
• Find proportion/probability, given the scores

15
Gamma Distribution

• A random variable Y is said to have a gamma
  distribution with parameters ?gt0 and ?0 if and
  only if the density function of Y is

16
Mean and Variance of Gamma Probability
Distribution

• Mean ?EY??
• Variance ?2VY??2
• MGF my(t)(1-?t)-?
• Special cases
• Chi-square distribution with ? degrees of freedom
  if ? ? /2 and ?2
• Exponential distribution when ? 1

17
Chi-square Probability Distribution

• A random variable Y is said to have a chi-square
  distribution with ? degrees of freedom if and
  only if the density function of Y is

Mean ? EY? , Variance ?2VY 2 ? MGF
my(t)(1-2t)-? /2
18
Exponential Probability Distribution

• A random variable Y is said to have an
  exponential distribution with parameter ?gt0 if
  and only if the density function of Y is

Mean ? EY? , Variance ?2 VY ?2 MGF
my(t)(1- ?t)-1 Memoryless property
P(YgtabYgta)P(Ygtb)
19
Homework

• p.173-175 4.46, 4.47, 4.54,  4.57, 4.63 p180
  4.68, 4.71, 4.72, 4.76

20
Beta Probability Distribution

• A random variable Y is said to have a beta
  probability distribution with parameters ?gt and
  ?gt0 if and only if the density function of Y is

21
Mean and Variance of Beta Y

• Mean ? EY ?/(??)
• Variance ?2 VY? ? /((??)2(??1))
• MGF no closed form
• Special cases ?? 1, Y is uniform distribution.
  
• That is f(y)1 , 0?y?1
• 0, elsewhere

22
Section 4.9

• Moment
• Moment of Function of Random Variables

23
Moment

• Moment If Y is a continuous random variable,
  then the kth moment about the origin is given by
• ?k EYk, k1,2,
• The kth moment about the mean, or the kth central
  moment, is given by
• ?k E(Y- ?)k, k1,2,

24
Moment Generating Function (MGF)

• MGF If Y is a continuous random variable, then
  the moment-generating function of Y is given by
• my(t)EetY
• The moment-generating function is said to exist
  if there exists a constant bgt0 such that my(t) is
  finite for t?b.
• The kth moment about origin is

25
MGF of Function

• Theorem Let Y be a random variable with density
  function f(y) and g(y) be a function of Y. then
  the moment-generating function for g(Y) is

26
Homework

• p185 4.91, 4.92p193-194 4.104, 4.108, 4.113

27
Section 4.10

• Tchebysheffs Theorem
• (Chebyshevs Inequality)
• Let Y be any random variable with finite mean
  ? and variance ?2. Then, for any kgt0, P(Y- ?
  lt k?) ? 1 (1/k2)or P(Y- ? ? k?) ? (1/k2)
• Compare it to 68-95-99.7 rules for normal
  distribution