Random variables

Let 5 be a sample space with probability distribution P.

2.11 Definition A random variable X is a function from the sample space 5 to the set of real numbers; to each simple event s, e 5, X assigns a real number X(s,).

Since 5 is assumed to be finite, X can only take on a finite number of values.

  • 2.12 Definition Let A be a random variable on 5. Iht expected value ox mean of X is E{X) = T,Si£SX(si)P(si)-
  • 2.13 Fact Let A be a random variable on S. Then E(X) = x ' = x)•
  • 2.14 Fact If Xi, X2,... , X,n are random variables on 5, and ai,a2,... , am are real numbers,

then E(YZi aiX>) = YZi aiE(xi)-

2.15 Definition The variance of a random variable X of mean p is a non-negative number defined by

The standard deviation of X is the non-negative square root of Var(X).

If a random variable has small variance then large deviations from the mean are unlikely to be observed. Tins statement is made more precise below.

2.16 Fact (Cliebyshev’s inequality) Let X be a random variable with mean p = E(X) and variance a2 = Var(X). Then for any t > 0,

Binomial distribution

  • 2.17 Definition Let n and к be non-negative integers. The binomial coefficient (£) is the number of different ways of choosing к distinct objects from a set of n distinct objects, where the order of choice is not important.
  • 2.18 Fact (properties of binomial coefficients) Let n and к be non-negative integers.

® (fc) = Щп-к)] ■

W (I) - („%)■

« о = Ю + «.)■

2.19 Fact (binomial theorem) For any real numbers a, b, and non-negative integer n, (a+b)n =

YX,o

  • 2.20 Definition A Bernoulli trial is an experiment with exactly two possible outcomes, called success and failure.
  • 2.21 Fact Suppose that the probability of success on a particular Bernoulli trial is p. Then the probability of exactly к successes in a sequence of n such independent trials is

  • 2.22 Definition The probability distribution (2.1) is called the binomial distribution.
  • 2.23 Fact The expected number of successes in a sequence of n independent Bernoulli trials, with probability p of success in each trial, is up. The variance of the number of successes is np(l — p).
  • 2.24 Fact (law of large numbers) Let X be the random variable denoting the fraction of successes in n independent Bernoulli trials, with probability p of success in each trial. Then for any e > 0,

In other words, as n gets larger, the proportion of successes should be close to p, the probability of success in each trial.

 
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