45-733 PROBABILITY AND STATISTICS I Notes #2B


January 2000



Continuous Random Variables

  1. Continuous Distributions
    P[X Î A] = òA f(x)dx
  2. Properties of Continuous Probability Distributions
    a) f(x) ³ 0 over the real line
    b) òf(x)dx = 1 over the real line
    c) P[a £ X £ b] = òab f(x)dx = F(b) - F(a)
  3. P(X = x) = 0. To see this:
    P(X = a) = òaa f(x)dx = F(a) - F(a) = 0
  4. Continuous Uniform Distribution
    
                             æ c  a < x < b
                      f(x) = ç
                             è 0 otherwise
    
    This is what the function looks like:

  5. To solve for the constant, c, use (b) in (22). That is:
    òab cdx = cx|ab = c[b-a] = 1; hence, c = 1/(b-a)
  6. Example: Let f(x) be:
    
                             æ c(1 - x3)  0 < x < 1
                      f(x) = ç
                             è   0 otherwise
    
    Find c.
    ò01 c(1 - x3)dx = c[x - x4/4]|01 = c[1 - 1/4] = c(3/4) = 1
    Hence, c = 4/3

    Find F(1/2).
    F(1/2) = P(X < 1/2) = ò01/2 (4/3)(1 - x3)dx = (4/3)[x - x4/4]|01/2 =
    (4/3)[1/2 - 1/64] = 31/48
  7. Example: Let f(x) be:
    
                             æ c/x  0 < x < 1
                      f(x) = ç
                             è 0 otherwise
    
    Find c.
    ò01 cxdx = clnx|01 = c[0 - - ¥] = ??
    So this is not a probability distribution.

  8. Problem 4.8 p.145
    
                             æ c(2 - y)  0 < y < 2
                      f(y) = ç
                             è 0 otherwise
    
    1. Find c
      ò02 c(2 - y)dy = c[2y - y2/2]|02 = c(4 - 4/2) = 2c
      Hence, c = 1/2

    2. Find F(y)
      F(y) = P(Y < y) = ò0y 1/2(2 - t)dt =
      1/2[2t - t2/2]|0y = 1/2(2y - y2/2) = y - y2/4
      . Hence:
      
                               æ   0   y < 0
                               ç
                        F(y) = ç  y - y2/4  0 £ y £ 2
                               ç 
                               è   1   y > 2