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Brief Review

Brief Review. Probability and Statistics. Probability distributions. Continuous distributions. Defn (density function). Let x denote a continuous random variable then f ( x ) is called the density function of x 1) f ( x ) ≥ 0 2) 3) .

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Brief Review

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  1. Brief Review Probability and Statistics

  2. Probability distributions Continuous distributions

  3. Defn (density function) Let x denote a continuous random variable then f(x) is called the density function of x 1) f(x) ≥ 0 2) 3)

  4. Examples of some important Univariate distributions

  5. 1.The Normal distribution A common probability density curve is the “Normal” density curve - symmetric and bell shaped Comment:If m = 0 and s = 1 the distribution is called the standard normal distribution Normal distribution with m = 50 and s =15 Normal distribution with m = 70 and s =20

  6. 2.The Chi-squared distribution with n degrees of freedom

  7. Comment: If z1, z2, ..., zn are independent random variables each having a standard normal distribution then U = has a chi-squared distribution with n degrees of freedom.

  8. 3. The F distribution withn1 degrees of freedom in the numerator and n2 degrees of freedom in the denominator if x  0 where K =

  9. Comment: If U1 and U2 are independent random variables each having Chi-squared distribution with n1 and n2 degrees of freedom respectively then F = has a F distribution with n1 degrees of freedom in the numerator and n2 degrees of freedom in the denominator

  10. 4.The t distribution with n degrees of freedom where K =

  11. Comment: If zand U are independent random variables, and z has a standard Normal distribution while U has a Chi-squared distribution with n degrees of freedom then t = has a t distribution with n degrees of freedom.

  12. Multivariate Distributions

  13. Defn (Joint density function) Let x = (x1 ,x2 ,x3 , ... , xn) denote a vector of continuous random variables then f(x) = f(x1 ,x2 ,x3 , ... , xn) is called the joint density function of x = (x1 ,x2 ,x3 , ... , xn) if 1) f(x) ≥ 0 2) 3)

  14. Note:

  15. Defn (Marginal density function) The marginal density of x1 = (x1 ,x2 ,x3 , ... , xp) (p < n) is defined by: f1(x1) = = where x2 = (xp+1 ,xp+2 ,xp+3 , ... , xn) The marginal density of x2 = (xp+1 ,xp+2 ,xp+3 , ... , xn) is defined by: f2(x2) = = where x1 = (x1 ,x2 ,x3 , ... , xp)

  16. Defn (Conditional density function) The conditional density of x1 given x2 (defined in previous slide) (p < n) is defined by: f1|2(x1 |x2) = conditional density of x2 given x1 is defined by: f2|1(x2 |x1)=

  17. Marginal densities describe how the subvector xi behaves ignoring xj Conditional densities describe how the subvector xi behaves when the subvector xj is held fixed

  18. Defn (Independence) The two sub-vectors (x1 and x2) are called independent if: f(x) = f(x1, x2) = f1(x1)f2(x2) = product of marginals or the conditional density of xi given xj : fi|j(xi |xj) = fi(xi) = marginal density of xi

  19. Example (p-variate Normal) The random vector x (p× 1) is said to have the p-variate Normal distribution with mean vector m(p× 1) and covariance matrix S(p×p) (a Positive definite matrix) (written x ~ Np(m,S)) if:

  20. Example (bivariate Normal) The random vector is said to have the bivariate Normal distribution with mean vector and covariance matrix

  21. The Univariate Normal distribution (mean m, Variance s2)

  22. Theorem (Transformations) Let x = (x1 ,x2 ,x3 , ... , xn) denote a vector of continuous random variables with joint density function f(x1 ,x2 ,x3 , ... , xn) = f(x). Let y1 =f1(x1 ,x2 ,x3 , ... , xn) y2 =f2(x1 ,x2 ,x3 , ... , xn) ... yn =fn(x1 ,x2 ,x3 , ... , xn) define a 1-1 transformation of x into y.

  23. Then the joint density of y is g(y) given by: g(y) = f(x)|J| where = the Jacobian of the transformation

  24. Corollary (Linear Transformations) Let x = (x1 ,x2 ,x3 , ... , xn) denote a vector of continuous random variables with joint density function f(x1 ,x2 ,x3 , ... , xn) = f(x). Let y1 = a11x1 + a12x2 + a13x3 , ... + a1nxn y2 = a21x1 + a22x2 + a23x3 , ... + a2nxn ... yn = an1x1 + an2x2 + an3x3 , ... + annxn define a 1-1 transformation of x into y.

  25. Then the joint density of y is g(y) given by:

  26. Corollary (Linear Transformations for Normal Random variables) Let x = (x1 ,x2 ,x3 , ... , xn) denote a vector of continuous random variables having an n-variate Normal distribution with mean vector m and covariance matrix S. i.e. x ~ Nn(m, S) Let y1 = a11x1 + a12x2 + a13x3 , ... + a1nxn y2 = a21x1 + a22x2 + a23x3 , ... + a2nxn ... yn = an1x1 + an2x2 + an3x3 , ... + annxn define a 1-1 transformation of x into y. Then y = (y1 ,y2 ,y3 , ... , yn) ~ Nn(Am,ASA')

  27. Defn (Expectation) Let x = (x1 ,x2 ,x3 , ... , xn) denote a vector of continuous random variables with joint density function f(x) = f(x1 ,x2 ,x3 , ... , xn). Let U = h(x)= h(x1 ,x2 ,x3 , ... , xn) Then

  28. Defn (Conditional Expectation) Let x = (x1 ,x2 ,x3 , ... , xn) = (x1 , x2 ) denote a vector of continuous random variables with joint density function f(x) = f(x1 ,x2 ,x3 , ... , xn) = f(x1 , x2 ). Let U = h(x1)= h(x1 ,x2 ,x3 , ... , xp) Then the conditional expectation of U given x2

  29. Defn (Variance) Let x = (x1 ,x2 ,x3 , ... , xn) denote a vector of continuous random variables with joint density function f(x) = f(x1 ,x2 ,x3 , ... , xn). Let U = h(x)= h(x1 ,x2 ,x3 , ... , xn) Then

  30. Defn (Conditional Variance) Let x = (x1 ,x2 ,x3 , ... , xn) = (x1 , x2 ) denote a vector of continuous random variables with joint density function f(x) = f(x1 ,x2 ,x3 , ... , xn) = f(x1 , x2 ). Let U = h(x1)= h(x1 ,x2 ,x3 , ... , xp) Then the conditional variance of U given x2

  31. Defn (Covariance, Correlation) Let x = (x1 ,x2 ,x3 , ... , xn) denote a vector of continuous random variables with joint density function f(x) = f(x1 ,x2 ,x3 , ... , xn). Let U = h(x)= h(x1 ,x2 ,x3 , ... , xn) and V = g(x)=g(x1 ,x2 ,x3 , ... , xn) Then the covariance of U and V.

  32. Properties • Expectation • Variance • Covariance • Correlation

  33. E[a1x1 + a2x2 + a3x3 + ... + anxn] = a1E[x1] + a2E[x2] + a3E[x3] + ... + anE[xn] or E[a'x] = a'E[x]

  34. E[UV] = E[h(x1)g(x2)] = E[U]E[V] = E[h(x1)]E[g(x2)] if x1 and x2 are independent

  35. Var[a1x1 + a2x2 + a3x3 + ... + anxn] or Var[a'x] = a′Sa

  36. Cov[a1x1 + a2x2 + ... + anxn , b1x1 + b2x2 + ... + bnxn] or Cov[a'x, b'x] = a′Sb

  37. Statistical Inference Making decisions from data

  38. There are two main areas of Statistical Inference • Estimation – deciding on the value of a parameter • Point estimation • Confidence Interval, Confidence region Estimation • Hypothesis testing • Deciding if a statement (hypotheisis) about a parameter is True or False

  39. The general statistical modelMost data fits this situation

  40. Defn (The Classical Statistical Model) The data vector x = (x1 ,x2 ,x3 , ... , xn) The model Let f(x|q) = f(x1 ,x2 , ... , xn|q1 , q2 ,... , qp) denote the joint density of the data vector x = (x1 ,x2 ,x3 , ... , xn) of observations where the unknown parameter vector qW (a subset of p-dimensional space).

  41. An Example The data vector x = (x1 ,x2 ,x3 , ... , xn) a sample from the normal distribution with mean m and variance s2 The model Then f(x|m , s2) = f(x1 ,x2 , ... , xn|m , s2), the joint density of x = (x1 ,x2 ,x3 , ... , xn) takes on the form: where the unknown parameter vector q = (m , s2) W ={(x,y)|-∞ < x < ∞ , 0 ≤ y < ∞}.

  42. Defn (Sufficient Statistics) Let x have joint density f(x|q) where the unknown parameter vector qW. Then S = (S1(x) ,S2(x) ,S3(x) , ... , Sk(x)) is called a set of sufficient statistics for the parameter vector q if the conditional distribution of x given S = (S1(x) ,S2(x) ,S3(x) , ... , Sk(x)) is not functionally dependent on the parameter vector q. A set of sufficient statistics contains all of the information concerning the unknown parameter vector

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