CS723 - Probability and Stochastic Processes

Lecture No. 09

In Previous Lectures Probability theoryEqually likely outcomesThoroughly analyzed the random partitioning of a unit interval with two randomly chosen pointsThe points were equally likely to fall on any point on unit interval

Equally Likely Outcomes Coin toss: two equally likely outcomes Dice roll: six equally likely outcomes Semi-cube roll: not equally likely, so mark the sides A, B, C, D, E, and F Sack with 6 green and 3 red balls: probability of 2 green balls =41.67% probability of 2 red balls = 8.33% Number of customers entering in a restaurant in a 10-minute interval.

Random Variables Transformation or mapping of outcomes to numbers on real line Probability maps values of RV to [0,1] Roll of two dice: X is random variable that represents the sum Pr(X=2) = Pr({(1,1)}) = f(2) = 1/36 f(4) = Pr(X=4) = Pr({(3,1),(2,2),(1,3)}) = 3/36

Probability Mass Function

Chuk-a-luck Three dice are thrown and you pick a number between 1 and 6 You loose one dollar if your number does not show up on any dice You win 3 dollars if your number shows up on all three dice X is your gain/loss and Δ = {-1, 1, 2, 3} f(3) = 1/216, f(2) = 15/216 = 5/72, f(1) = 75/216 = 25/72 & f(-1)= 125/216

Binomial Distribution Sample space with only two outcomes success and failure Pr(success) = p & Pr(failure) = (1-p) Repeat the experiment N times under same conditions (with replacement) Total number of outcomes in sample space is 2 N (words from alphabets) Random variable is X = No. of successes f(k) = Pr(X=k) = (N,k) p k (1-p) N-k

People Entering KFC When a person enter the franchise, he enters instantaneously People enter the franchise independently of each other The time is divided in 5 secs intervals

Poisson Distribution Probability of success VERY small and number of repetitions VERY large The random variable can take any value that is a positive integer (or zero) PMF is given by f(k) = Pr(X=k) = e -λ (λ k /k!)

Cumulative Distribution Function CDF is continuous on the real line and is probability of a complex event F(t) = Pr(-∞ < X ≤ t) = ∑ f(a) s.t. a ≤ t, t ε R lim t → -∞ F(t) = 0 and lim t → ∞ F(t) = 1 f(t) = F(t) – lim s → t F(s)

Important Points A random variable is a function from sample space to real line Finite sample spaces always give discrete random variables Probabilities are assigned to events in terms of values of random variable We can work with a PMF without knowing the underlying experiment Sum of all values of PMF should be 1