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Title: Today


1
Today
  • Today More Chapter 5
  • Reading
  • Important Sections in Chapter 5 5.1-5.11
  • Only material covered in class
  • Note we have not, and will not cover
    moment/probability generating functions
  • Suggested problems 5.1, 5.2, 5.3, 5.15, 5.25,
    5.33, 5.38, 5.47, 5.53, 5.62
  • Exam will be returned in Discussion Session
  • Important Sections in Chapter 5

2
Example
  • Suppose X and Y have joint pdf f(x,y)xy for
    0ltxlt1, 0ltylt1
  • Find the marginal distributions of X and Y
  • Is there a linear relationship between X and Y?

3
Covariance and Correlation
  • Recall, the covariance between tow random
    variables is
  • The covariance is

4
Properties
  • Cov(X,Y)E(XY)-µXµY
  • Cov(X,Y)Cov(Y,X)
  • Cov(aX,bY)abCov(X,Y)
  • Cov(XY,Z)Cov(X,Z)Cov(Y,Z)

5
Example
  • Suppose X and Y have joint pdf f(x,y)xy for
    0ltxlt1, 0ltylt1
  • Is there a linear relationship between X and Y?
  • What is Cov(3X,-4Y)?
  • What is the correlation between 3X and -4Y

6
Independence
  • In the discrete case, two random variables are
    independent if
  • In the continuous case, X and Y are independent
    if
  • If two random variables are independent, their
    correlation (covariance) is

7
Example
  • Suppose X and Y have joint pdf f(x,y)xy for
    0ltxlt1, 0ltylt1
  • Are X and Y independent?

8
Example
  • Suppose X and Y have joint pdf f(x,y)xy for
    0ltxlt1, 0ltylt1
  • Are X and Y independent?

9
Hard Example
  • Suppose X and Y have joint pdf f(x,y)45x2y2 for
    xylt1
  • Are X and Y independent?
  • What is their covariance?

10
Conditional Distributions
  • Similar to the discrete case, we can update our
    probability function if one of the random
    variables has been observed
  • In the discrete case, the conditional probability
    function is
  • In the continuous case, the conditional pdf is

11
Example
  • Suppose X and Y have joint pdf f(x,y)xy for
    0ltxlt1, 0ltylt1
  • What is the conditional distribution of X given
    Yy?
  • Find the probability that Xlt1/2 given Y1/2

12
Normal Distribution
  • One of the most important distributions is the
    Normal distribution
  • This is the famous bell shaped distribution
  • The pdf of the normal distribution is
  • Where the mean and variance are

13
Normal Distribution
  • A common reference distribution (as we shall see
    later in the course) is the standard normal
    distribution, which has mean 0 and variance of 1
  • The pdf of the standard normal is
  • Note, we denote the standard normal random
    variable by Z

14
CDF of the Normal Distribution
  • The cdf of a continuous random variable,Z, is
    F(z)P(Zltz)
  • For the standard normal distribution this is

15
Relating the Standard Normal to Other Normal
Distributions
  • Can use the standard normal distribution to help
    compute probabilities from other normal
    distributions
  • This can be done using a z-score
  • A random variable X with mean µ and variance s
    has a normal distribution only if the z-score has
    a standard normal

16
Relating the Standard Normal to Other Normal
Distributions
  • If the z-score has a standard normal
    distribution, can use the standard normal to
    compute probabilities
  • Table II gives values for the cdf of the standard
    normal

17
Example
  • The height of female students at a University
    follows a normal distribution with mean of 65
    inches and standard deviation of 2 inches
  • Find the probability that a randomly selected
    female student has a height less than 58 inches
  • What is the 99th percentile of this distribution?

18
Finding a Percentile
  • Can use the relationship between Z and the random
    variable X to compute percentiles for the
    distribution of X
  • The 100pth percentile of normally distributed
    random variable X with mean µ and variance s can
    be found using the standard normal distribution

19
Example
  • The height of female students at a University
    follows a normal distribution with mean of 65
    inches and standard deviation of 2 inches
  • What is the 99th percentile of this distribution?
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