1   Exam 1

next Thurs Feb 20.

one 2-sided crib sheet allowed.

2   Notation

How to parse \(F_X(x)\)

1. Uppercase F means that this is a cdf. Different letters may indicate different distributions.
2. The subscript X is the name of the random variable.
3. The x is an argument, i.e., an input.
4. \(F_X(x)\) returns the probability that the random variable is less or equal to the value x, i.e. prob(X<=x).

3   Matlab

1. Matlab, Mathematica, and Maple all will help you do problems too big to do by hand. Sometime I'll demo one or the other.

2. Matlab

   1. Major functions:

      cdf(dist,X,A,...)
      pdf(dist,X,A,...)
      

   2. Common cases of dist (there are many others):

      'Binomial'
      'Exponential'
      'Poisson'
      'Normal'
      'Geometric'
      'Uniform'
      'Discrete Uniform'
      

   3. Examples:

      pdf('Normal',-2:2,0,1)
      cdf('Normal',-2:2,0,1)
      
      p=0.2
      n=10
      k=0:10
      bp=pdf('Binomial',k,n,p)
      bar(k,bp)
      grid on
      
      bc=cdf('Binomial',k,n,p)
      bar(k,bc)
      grid on
      
      x=-3:.2:3
      np=pdf('Normal',x,0,1)
      plot(x,np)
      

   4. Interactive GUI to explore distributions: disttool

   5. Random numbers:

      rand(3)
      rand(1,5)
      randn(1,10)
      randn(1,10)*100+500
      randi(100,4)
      

   6. Interactive GUI to explore random numbers: randtool

   7. Plotting two things at once:

      x=-3:.2:3
      n1=pdf('Normal',x,0,1)
      n2=pdf('Normal',x,0,2)
      plot(x,n1,n2)
      plot(x,n1,x,n2)
      plot(x,n1,'--r',x,n2,'.g')
      

3. Use Matlab to compute a geometric pdf w/o using the builtin function.

4. Review. Which of the following do you prefer to use?

   1. Matlab
   2. Maple
   3. Mathematica
   4. Paper. It was good enough for Bernoulli and Gauss; it's good enough for me.
   5. Something else (please email about it me after the class).

4   Chapter 4 ctd

1. Text 4.2 p 148 pdf

2. Simple continuous r.v. examples: uniform, exponential.

3. The exponential distribution complements the Poisson distribution. The Poisson describes the number of arrivals per unit time. The exponential describes the distribution of the times between consecutive arrivals.

   The exponential is the continuous analog to the geometric. If the random variable is the integral number of seconds, use geometric. If the r.v. is the real number time, use exponential.

   Ex 4.7 p 150: exponential r.v.

4. Properties

   1. Memoryless.
   2. \(f(x) = \lambda e^{-\lambda x}\) if \(x\ge0\), 0 otherwise.
   3. Example: time for a radioactive atom to decay.

5. Skip 4.2.1 for now.

6. The most common continuous distribution is the normal distribution.

7. 4.2.2 p 152. Conditional probabilities work the same with continuous distributions as with discrete distributions.

8. p 154. Gaussian r.v.

   1. \(f(x) = \frac{1}{\sqrt{2\pi} \cdot \sigma} e^{\frac{-(x-\mu)^2}{2\sigma^2}}\)
   2. cdf often called \(\Psi(x)\)
   3. cdf complement:
      1. \(Q(x)=1-\Psi(x) = \int_x^\infty \frac{1}{\sqrt{2\pi} \cdot \sigma} e^{\frac{-(t-\mu)^2}{2\sigma^2}} dt\)
      2. E.g., if \(\mu=500, \sigma=100\),
         1. P[x>400]=0.66
         2. P[x>500]=0.5
         3. P[x>600]=0.16
         4. P[x>700]=0.02
         5. P[x>800]=0.001

9. Text 4.3 p 156 Expected value

10. Skip the other distributions (for now?).

5   Comic

Broomhilda