Random Variables and Probability Distributions
Random Variables and Probability Distributions ENGR 0020: Probability and statistics for Engineers I
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This 2 page One Day of Notes was uploaded by Emily Binakonsky on Friday January 23, 2015. The One Day of Notes belongs to ENGR 0020: Probability and statistics for Engineers I at University of Pittsburgh taught by Maryam Mofrad in Spring2015. Since its upload, it has received 202 views. For similar materials see Probability and Statistics for Engineers 1 in Engineering and Tech at University of Pittsburgh.
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Date Created: 01/23/15
Random Variables amp Probability Distribution 12115 Concept of a Random Variable o For a given sample space S of some experiment a random variable is any rule that associates a number with each outcome in S 0 Ex Toss dice number of defects in a 100 product sample ph of chemical compound number of people who vote for a specific candidate weight measurements of steel coil samples length of steel beams etc Discrete Sample Space is a sample that contains a finite number of possibilities or an gt unending sequence with as many elements as there are whole numbers B Continuous Sample Space is a sample space that contains an infinite number of possibilities equal to the number on points on a line segment C Discrete Random Variable is a random variable whose possible values either constitute a finite set or else can be listed in an infinite sequence in which there is a first element a second element etc and so on to countable infinite D Continuous Random Variable is if both of the following conditions apply 1 Its set of possible values consists either of all numbers in a single interval on the number line possibly infinite in extent eg from to co or all numbers in a disjoint union of such intervals eg 010 U 2535 2 No possible value of the variable has positive probability that is P X c 0 for any possible value c Ex Are the following random variable s discrete of continuous 1 The total number of points scored in a football game Discrete 2 The shelf life of a particular drug Continuous decreasing linear function 3 The length of a twoyear old black bass Continuous because the distribution of the whole population would be a smooth function The shape of the distribution would be similar to that of a bell curve 4 The number of Aircraft nearcollisions last year Discrete It d be like a step function ie you either have a nearcollision or you don t you can t get half a near collision nor 15 nearcollisions Probability Distribution The probability distribution for a random variable X tells us how the total probability of 1 for the sample space S is distributed among each of the mutually exclusive simple events or outcomes for X that describe the sample space SIMPLE Coin Coin PE1 X EVENT 1 2 E1 H E2 H E3 T E4 T H T H T Random Variables amp Probability Distribution 12115 Discrete Probability Distribution The set of ordered pairs x fx is a probability function probability mass function or probability distribution of the discrete random variable X if for each possible outcome x 1fx20 Zfox1 3PXxfx The Cumulative Distribution Function Discrete RV The cumulative distribution function Fx of a discrete random variable X with probability distribution fx is Fx P x s x 2m fx for to co Proposition For any two numbers a and b with a S b pa s X Sb Fb Fa quota represents the largest possible X value that is strictly less than a Note for integers as possible values for x pa s X sbFb Fa l