Sept. 28 - Oct. 2
Sept. 28 - Oct. 2 Stat 190
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This 5 page Class Notes was uploaded by Sarah Davis on Friday October 2, 2015. The Class Notes belongs to Stat 190 at Truman State University taught by Scott Thatcher in Fall 2015. Since its upload, it has received 55 views. For similar materials see Basic Statistics in Statistics at Truman State University.
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Date Created: 10/02/15
Book notes Ch 61 Discrete Random Variables Random variable a numerical measure of the outcome of a probability experiment so its value is determined by chance Random variables are typically denoted using capital letters such as X Discrete random variable has either nite or countable number of values The values of a discrete random variable can be plotted on a number line with space between each point Continuous random variable has in nite many values The values of a continuous random variable can be plotted on a line in an uninterrupted fashion Identify Discrete Probability Distributions Probability distribution of a discrete random variable X provides the possible values of the random variable and their corresponding probabilities A probability distribution can be in the form of a table graph or mathematical formula Rules for discrete probability distribution Let P denote the probability that the random variable X equals X then 1 2PXl X possible values of X 2 0 S PX S In a probability histogram the horizontal aXis corresponds to the value of the random variable and the vertical aXis represents the probability of each value of the random variable The mean of a discrete random variable u formula ux 2X PX Where X is the value of the random variable and PX is the probability of observing the value X Interpretation of the mean u Suppose an experiment is repeated n independent times and the value of the random variable X is recorded As the number of repetitions of the experiment increases the mean value of the n trials will approach ux the mean of the random variable X In other words let xl be the value of the random variable X after the rst experiment x2 be the value of the random variable X after the second experiment and so on Then mean x with a line over it xlx2xn n The difference between mean and u gets closer to 0 as n increases Mean is also called expected value and denoted with EX Standard deviation of a discrete random variable the standard deviation of a discrete random variable X is given by ox sq root 2xux2 Px sq root 2x2 Px u 2 x where x is the value of the random variable ux is the mean of the random variable and Px is the probability of observing a value of the random variable Sorry there39s no other way I can see to enter some of those math thingsie Square root symbols Sept 29 2015 inclass Random variable numeric variable whose value depends on the results of chance Discrete Random variable whose values come from a discrete set of numbers Usually comes from counting has integer values Example number of siblings Continuous can take on a range of real numbers Comes from measurement Example diamond weighs 92 ounces or 92353 and so on Probability distribution lists all the values of a random ariable and the probability of each Sometimes we39re given ones representing real data Experiment rolling two dice Variable sum of the 2 dice rolls X sum Probability continued 2 136 7 636 12 136 3 236 8 536 4 336 9 436 5 436 10 336 6 536 11 236 XYZetc represent random variables Xyzetc represent specific values of a random variable ie PX3236 XX means the event that random X is equal to X PXX means the probability that random variable X takes on value X x3 the event that the sum of 2 dice3 Mean of random variable If you were to perform an experiment many times what39s the mean of all the measurements Mean of random variable X is uX MX 2XPXX multiply each value of the random variable by its probability Sum the numbers XPXX 2l36236 Reach 12 add up the sums You get 252 25236 IS 32 3 66 3 6 YOUR ANSWER 4336l236etc Class Oct 1 2015 Binomial eXperiment random process with two outcomes success or failure n number of identical trials each trial is independent For a single trial Psuccess p and p is constant Pfailure lp If X the number of successes out of n trials we want to nd the formula for PXX Binomial Probability Formula gives probability of X successes in n trials PXX ways to have X probability of probability of nX successes in n X successes failures trials PXX nCXpAX1pAnX n nnln2n3 until the number reaches 1 ol 4C2 4 49 g 4C26 214 2 2x12x1 2 sample is taken from a nite population Xnumber in sample with a speci ed attribute Sample with replacement distribution of X is binomial Sample without replacement distribution of X is approximately binomial 0 If sample lt 5 then n lt 05 N We had a problem in which students from the class said whether they were for or against the smoking ban Out of 22 students 19 said yes and 3 said no The probability of getting a yes from a student is 75 and getting a no is 25 according to this data Sample of 22 probability of 19 success n22 Xl9 p75 PXl9 22Cl9 75Al925 3 22Cl9 in calculator I have a TI84 is 22 mathPRB hit nCr and then 19 nal power of 3 comes from 2219 PXl9 101 101 chance of getting a sample like our class Probability of getting at most 6 successes yes answers Pat most 6 PXS 6 PX S 6 PX6 PX5 PX4 0544 lt 5 unlikely gt5 not unlikely He put a document on Blackboard for us to use it39s a table of the binomial probability answers so we don39t have to longhand them if we don39t have calculators Longhand problem from earlier 22 X 21 X 20 X19 X 18 22Cl9 19X 18X l73X2Xl Since both topbottom have l9l cancel those and you39re left with 22 X 21 X 20 3X2Xl Book 62 Criteria for a binomial probability experiment 1 The experiment is performed a xed number of times Each repetition of the experiment is called a trial 2 The trials are independent the outcome of one trial will not affect the outcome of the others 3 For each trial there are two mutually exclusive disjoint outcomes success or failure 4 The probability of success is the same for each trial of the experiment Let X the number of successes in n trials of a binomial experiment Then X is called a binomial random variable Notation used in the Binomial probability distribution There are n independent trials of the experiment Let p denote the probability of success for each trial so that lp is the probability of failure for each trial Let X denote the number of successes in n independent trials of the experiment 0 O S x S n Binomial probability distribution function the probability of obtaining x successes in n independent trials of a binomial experiment is given by Px anprlpAnx X 0 l 2 n p is the probability of success Mean or expected value and standard deviation of a binomial random variable a binomial experiment with n independent trials and probability of success p has a mean and standard deviation given by the formulas ux np and G 2 sq rt npl39P For a xed p as the number of trials n in a binomial experiment increases the probability distribution of the random variable X becomes bellshaped As a rule of thumb if nplp Z 10 the probability distribution will be approximately bellshaped
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