distributions

The Binomial Distribution If a discrete random variable X has the following probability density function (p.d.f.), it is said to have a binomial distribution: P(X = x) = nCx q(n-x)px, where q = 1 - p p can be considered as the probability of a success, and q the probability of a failure. Note: nCr (“n choose r”) is more commonly written It means the number of ways of choosing r objects from a collection of n objects (see permutations and combinations). If a random variable X has a binomial distribution, we write X ~ B(n, p) (~ means ‘has distribution…’). n and p are known as the parameters of the distribution (n can be any integer greater than 0 and p can be any number between 0 and 1). All random variables with a binomial distribution have the above p.d.f., but may have different parameters (different values for n and p). Example A coin is thrown 10 times. Find the probability density function for X, where X is the random variable representing the number of heads obtained. The probability of throwing a head is ½ and the probability of throwing a tail is ½. Therefore, the probability of throwing 8 tails is (½)8 If we throw 2 heads and 8 tails, we could have thrown them HTTTTTHTT, or TTHTHTTTTT, or in a number of other ways. In fact, the total number of ways of throwing 2 heads and 8 tails is Hence the probability of throwing 2 heads and 8 tails is: 10C2 × (½)2 × (½)8 The Normal Distribution Standardizing normal random variables It is possible to relate all normal random variables to the standard normal. For example if X is normal with mean μ and variance σ2, then has mean zero and unit variance, that is Z has the standard normal distribution. Conversely, having a standard normal random variable Z we can always construct another normal random variable with specific mean μ and variance σ2: