# 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}£S^{X}(^{s}i)P(si)- - 2.13 Fact Let A be a random variable on
*S.*Then*E(X)*='^{x}^{= x})• - 2.14 Fact If
*Xi, X*are random variables on 5, and_{2},... , X,_{n}*ai,a*are real numbers,_{2},... , a_{m}

then *E(YZi ^{a}i^{X}>) = YZi ^{a}iE(^{x}i)-*

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 *a ^{2} =* 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.