PROBABILITY I M 362K
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This 3 page Class Notes was uploaded by Reyes Glover on Sunday September 6, 2015. The Class Notes belongs to M 362K at University of Texas at Austin taught by Staff in Fall. Since its upload, it has received 4 views. For similar materials see /class/181473/m-362k-university-of-texas-at-austin in Mathematics (M) at University of Texas at Austin.
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Date Created: 09/06/15
PROBABILITY I Test 2 review topics August 5 2008 1 Random variables a Random variables real valued functions X on the sample space S ie X S 7 R Cumulative distribution function is de ned as FxPX 7ooltxltoo 2 Discrete random variables Random variables X that can take at most countably many values xi 239 1 2 a Probability mass function is de ned as pm 7 PX 7 z b Normalization condition 1 c Cumulative distribution function Z migm Expectation expected value of X is EX Variance of X is VarX EX2 7 EX2 Expectation expected value of a function g of a random variable X is EgX Z The Bernoulli random variable with a parameter p X can take values 1 success with the probability p or 0 failure with the probability 1 7 p h The binomial random variable with parameters np X is the number of successes when n independent trials each of which can result in a success with the probability p are performed pk 2311 7p k Expected value of X EX np Variance of X VarX np1 7 p The Poisson random variable with a parameter A A Mk Pk5 E7 k0 00 Expected value of X EX A Variance of X Va le A j The binomial random variable with the parameters 7110 approaches Poisson with parameter A 71p7 if n a 007 p a 07 and np is of moderate size k The distribution of the number of events that occur in an interval of time t if 0 the probability that an event occurs in a small interval of time h is Ah small 7 where A gt 0 is xed 0 the probability that two or more events occur in a small interval of time h is small 0 events occur independently of each other is Poisson with the parameter At7 ie Wk kl 7 pk 6 k 0 oo 7 l The geometric random variable with a parameter p Suppose that independent trials are performed and that each trial can result in a success with probability p and failure with probability 1 7 p X is the number of trials performed until a success occurs Then7 X is a geometric random variable with the probability mass function Mk 1 7 MHZ Expected value of X ElX Variance of X VarX k12 1 17239 3 Continuous random variables Random variables X for which there exists a nonnegative function f R a R probability density such that PX E B fx dz for any subset B C R B This implies PX x 07 for any x Normalization condition f dx 1 Cumulative distribution function fix fy dy This implies f FQ Expectation expected value of X is ElX ff fx dz Variance of X is VarX ElXZ 7 EX2 Expectation expected value of a function g of a random variable X is EgX The uniform random variable X ffooogltgtfltgt d96 ifaltzlt 0 otherwise Expected value of X ElX Variance of X VarX The normal random variable X with parameters M and U 1 PM we r a 7 27m 700ltltOO Expected value of X ElX M Variance of X VarX 02 h If X is a normal random variable with the expectation M and variance 02 then Z X771 is a A A A L V normal random variable with the expectation 0 and variance 1 Cumulative distribution function of Z is denoted by 1 1 ltIgt e 2dy One has 7s 17 The De Moivre Laplace limit theorem The binomial random variable with the parameters 7110 approaches the normal distribution with expectation M mp7 and variance 02 71101710 when n a 00 This is a good approximation when np1 7 p is large In order to compare a discrete and a continuous distribution7 one needs to apply the continuity correction rst The exponential random variable X with the parameter A gt 0 Ae if z 2 07 fa T 0 otherwise Expected value of X ElX Variance of X VarX A nonnegative random variable X is memoryless if PXgtsthgttPXgts7 for all 57t20 An exponentially distributed random variable is memoryless and it is the unique distribution satisfying this property Distribution of a function of a random variable To nd the probability density fy of a random variable Y gX7 where X is a continuous random variable with the probability density fX7 nd the cumulative distribution function Fyy rst7 then differentiate If g is a strictly monotone function7 then fyy fx9 1yl 9 1yl ify 996 for some 96 0 ify 31 g for all x
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