The lecture Random Variables, Expected Values and Variance by Edu Pristine is from the course ARCHIV Probability Theory. It contains the following chapters:
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... distribution, the Bernoulli distribution, the binomial distribution, the geometric distribution, and the negative binomial distribution all uses discrete random variable. Examples: The number of brown eyed children in a family containing two children is a discretevariable as its set of possible values is a finite, countable set. The set contains the values, {0,1,2} , that is either in a family none of the child has a brown eye, either one of the child has a brown eye or both children have a brown eye ...
... The variable is said to be a continuous random variable if it can assume any value insome interval. Examples: distance, speed, time, etc.The probability it takes at any point is assumed to be in case of continuous Random Variable The normal distribution, continuous uniform distribution, Beta distribution, and Gamm a distribution are all continuous distributions. Useful when variable holds value between two limits. P[a ... number of jacks? In the above example the number of jacks can be taken as a random variable, X. Clearly X can take values 0 (no jack out of two cards drawn), 1(one jack), 2 ( both cards draw n are jack) P(X=0) = P(non-jack and non-jack) = P(non-jack)*P(non-jack) = 48/52 * 48 /52 = 144/169 P(X=1) = P(jack and non jack or non-jack or jack) = P(jack and non-jack) + P(non-jack and jack) = 4/52*48 /52 + 48/52*4/52 = 24/169 P(X=2) = 4/52 * 4/52 = 1/169 ... ... continuous variable found using integration over a limit at a particular value the probability measured is zero Hence P(x1 "d X "d x2) = P(x1 < X "d x2 ) = P( x1 "d X < x2) " Example: The PDF of a continuous variable is given by 6x 5 and 0 "d x "d 1 What is the probability that 0 < X < 0.5, the probability that X lies between 0 and 0.5. Cumulative probability distribution. A cumulative probability density function for a r.v. is defined ... ... distribution are stable, then the distribution itsel f is said to be a stable distribution. If X 1and X 2are the independent random variables from a stable distribution, and a, b are scalar variables, then the variable Y = a.X 1+ b.X 2also follows the same distribution. The Expected value of Y: E(Y) = a.E(X1) + b.E(X 2) " The Variance of Y: Var(Y)= a2 .Var(X 1) + b 2 .Var(X 2) ... ... Knowledge Management Pristine7 Scalar multiplication of a random number. On multiplication of a discrete random variable by a fixed number, the probabilities remain unchanged but the possibilities are multiplied by that fixed number. Example: Suppose X is a discrete random variable, then 2X is defined by the same probabilitydistribution as , that ... ... probabilit xof value. Expected value E[X] = If the probabilities of obtaining the amounts a 1, a 2, a 3, &,a k, are p 1, p 2, p 3, &,p k. Then the expected value is E = a 1* p 1+ a 2* p 2+ a 3* p 3+ &.+ a k* p k. Example: A drug is used to maintain a steady rate in patients who have suffered a mil d heart attack. Let X denote the number of heartbeats per minute obtained per ... ... X = ? 2 = E[X 2 ] (E[X]) 2 Standard Deviatio. The positive root of variance Not all random variables have a standard deviation, since these expected values need not exist If the variable is continuous, in that case we replace summation with integration Example: Find the variance and standard deviation for previous example E[X] = 35, E[X 2 ] = 1231.6 Variance = 1231.6 35*35 = ..