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Unformatted text preview: Stat 225 Lecture Notes Named Discrete Distributions Ryan Martin Spring 2008 1 Bernoulli Trials and Binomial RVs There are several popular categories of random variables and we will study each of these categories in detail. We begin with the simplest. In many applications, a random experiment is performed and whether or not a par- ticular event occurs is recorded. For example, we flip a coin and observe whether the coin lands on H or T. In this context, occurrence of the event is called a success and non-occurence is called a failure (in the coin flipping example, H = success and T = failure). Generally p denotes the probability of success on each repitition and q = 1 p denotes the probability of failure. Consider the random variable X , given by X = braceleftBigg 1 if the experiment is a success if the experiment is a failure If p is the probability of success, then this random variable is called a Bernoulli (or indicator ) random variable with parameter p . This is denoted by X Ber( p ). Some simple examples of Bernoulli random variables are: Coin tossing: X = 1 if toss is H, X = 0 if toss is T. Betting on red in roulette: X = 1 if ball lands on a red, X = 0 if not. Drug Effectiveness: X = 1 if the patients health improves, X = 0 if not. Exercise 1.1. If X Ber( p ), write down its PMF. Check that it sums to 1. Exercise 1.2. If X Ber( p ), show that E ( X ) = p and V ( X ) = p (1 p ). Definition 1.3. Repeated trials of a random experiment are called Bernoulli trials if (i) the trials are independent of one another; (ii) the result of each trial is classified as success or failure; (iii) the success probability remains the same from trial to trial. 1 1 2 3 4 5 6 0.0 0.1 0.2 0.3 0.4 0.5 X ~ Bin( 6 , 0.1 ) x p(x) 1 2 3 4 5 6 0.05 0.10 0.15 0.20 0.25 0.30 X ~ Bin( 6 , 0.5 ) x p(x) 1 2 3 4 5 6 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 X ~ Bin( 6 , 0.75 ) x p(x) Fig 1: Plot of Binomial PMF for n = 6 and p = 0 . 1 , . 5 , . 75 Exercise 1.4. Determine whether or not the following scenarios satisfy the definintion of Bernoulli trials (a) Play 10 games of poker and for each game observe win or lose. (b) Draw 5 balls without replacement from an urn with 10 balls (5 white, 5 red) and observe red or not. (c) Watch the weather for a week and each day observe whether or not it rains. In a sequence of Bernoulli trials, it might be of interest to count the number of successes. For example, if you play the lottery 10 times and count how many times you win. The number of successes X in a sequence of n Bernoulli trials, each with success probability p , is called a Binomial random variable with parameters n and p , and is denoted by X Bin( n, p ). Binomial is one of the most important classes of RVs....
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