2 CombinatoricsNotation: Population size Subpopulation size Ordered SampleHow many samples of size r can be formed from a population of size n?Sampling with replacement and orderingSampling without replacement and with ordering
3 How many samples of size r can be formed from a population of size n? Sampling without replacement and without orderingSampling with replacement and without ordering
4 Bernoulli TrialsIndependent trials that result in a success with probability p any failure probability 1-p.
5 Conditional Probabilities Given two events, E1 & E2, defined on the same probability space with corresponding probabilities P(E1) & P(E2):
6 Example 4.5An information source produces 0 and 1 with probabilities 0.3 and 0.7, respectively. The output of the source is transmitted via a channel that has a probability of error (turning a 1 into a 0 or a 0 into a 1) of 0.2.What is the probability that at the output a 1 is observed?What is the probability that a 1 was the output of the source if at the output of the channel a 1 is observed?
7 Random Variables Working with Sets has its limitations Selecting events in ΩAssigning P[ ]Verification of 3 AxiomsWould like to leave set theory and move into more advanced & widely used mathematics(i.e., Integration, Derivatives, Limits…)Random Variables:Maps sets in Ω to RSubsets of the real line of the form are called Borel sets. A collection of Borel sets is called a Borel σ-fields.A function that will associate events in Ω with the Borel sets is called a Random Variable.
8 Cumulative Distribution Function (CDF) CDF of a random variable X is defined as: or Properties:
9 Probability Density Function (PDF) PDF of a random variable X is defined as: Properties: For discrete random variables, this is known as the Probability Mass Function (PMF)
10 Example 4.6A coin is flipped three times and the random variable X denotes the total number of heads that show up. The probability of a head in one flip of this coin is denoted by p.What values can the random variable X take?What is the PMF of the random variable X?Derive and plot the CDF of X.What is the probability that X exceeds 1?
11 Uniform Random Variable This a continuous random variable taking values between a and b with equal probabilities over intervals of equal length.
12 Bernoulli Random Variable Only two outcomes (e.g., Heads or Tails) Probability Mass Function: Leads to binomial law (sampling w/o replacement & w/o ordering) Example: 10 independent, binary pulses per second arrive at a receiver. The error probability (that is, a zero received as a one or vice versa) is What is the probability of at least one error/second?
13 Binomial Law ExampleFive missiles are fired against an aircraft carrier in the ocean. It takes at least two direct hits to sink the carrier. All five missiles are on the correct trajectory must get through the “point defense” guns of the carrier. It is known that the point defense guns can destroy a missile with probability P = 0.9. What is the probability that the carrier will still be afloat after the encounter?
14 Uniform DistributionProbability Density Function (pdf) Cumulative distribution function A resistor r is an RV uniform distribution between 900 and 1100 ohms. Find the probability that r is between 950 and 1050 ohms.
15 Gaussian (Normal) Random Variable A continuous random variable described by the density function: Properties:
16 Special Case: CDF for N(0,1) The CDF for the normalized Gaussian random variable with m = 0 and σ = 1 is: Pre-Normalized Gaussian: Another related function (complimentary error function) used for finding P(X > x) is: Properties:
17 Gaussian ExampleA random variable is N(1000; 2500). Find the probability that x is between 900 and 1050.
19 The Central Limit Theorem Let be a set of random variables with the following properties:The Xk with k = 1, 2, …, n are statistically independentThe Xk all have the same probability density functionBoth the mean and the variance exist for each XkLet Y be a new random variable defined asThe, according to the central limit theorem, the normalized random variableApproaches a Gaussian random variable with zero mean and unit variance as the number of random variablesIncreases without limit.
20 Functions of Random Variables Let X be a r.v. with known CDF and PDF. If g(X) is a function of the r.v. X thenExample:Given the function where a > 0 and b are constants and X is a r.v. withFind
21 Statistical AveragesThe expected value of the random variable X is defined as Special Cases: Characteristic Functions:
22 Multiple Random Variables The joint CDF of X and Y is and its joint PDF is Properties:
23 Expected ValuesGiven g(X,Y) as a function of X and Y, the expected value isSpecial Cases:Correlation of X & YCovariance of X & YConditional PDFX and Y are statistically independent if:
24 Example Two random variables X and Y are distributed according to Find the value of the constant K.Find the marginal density functions of X and Y .Are X and Y independent?
26 Jointly Gaussian R.V.’sDefinition: X and Y are jointly Gaussian if where ρ is the correlation coefficient between X and Y. If ρ = 0, then If jointly Gaussian, then the following are also Gaussian
27 n - Jointly Gaussian R.V.’s Definition: is jointly Gaussian if where and
28 Jointly Gaussian Properties Any subset of is a vector of jointly Gaussian R.V.’sJointly Gaussian R.V.’s are completely characterized byAny collection of R.V.’s are uncorrelated iff is diagonal. Also, independence implies their non-correlation. For jointly Gaussian R.V.’s,A collection of uncorrelated R.V.’s, each of which is Gaussian, may not be jointly Gaussian.If X is jointly Gaussian, then is also jointly Gaussian with:
29 ExampleLet A be a binary random variable that takes the values of +1 and -1 with equal probabilities. Let A and X are statistically independent. Let Y = A X.Find the pdfFind the covariance cov(X,Y).Find the covariance cov(X2,Y2).Are X and Y jointly Gaussian?