Discrete Random Variables

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4 Properties of a Binomial Expansion

Fixed number of trials (n)

Two possible outcomes in a trial (success or failure)

Trials are independent

Probability of success (p) remains constant

If X is the number of successes in n trials, X~B(n,p)

Formula of Binomial Distribution

Makes sense if you think of it in terms of tree diagrams and the different ways in which you can achieve a particular outcome

If X~B(n,p) then P(X=r) = ncr p^r q^n-r

note nc1 = n

Tables

P(X=n) = P(X=n) - P(X=n-1)

P(XP(X>n) = 1 - P(X=n)

P(>/=n) = 1 - P(X=n-1)

If p> 0.5, then use probability of failure (call variable 'Y') and know that the number of failures corresponds to the difference betwwen the number of trials and the number of successes. Thus, when X~B (5, 0.9) let Y~B (5, 0.1). When Y = 2, X will then equal 3. So P(X=3) = P(Y>/=2)

Mean and Variance

If X ~ B(n, p)

mean = μ = E(X) = np

variance = σ² = npq where q = 1 - p (also known as the probability of failure)

let x be the random variable number of faulty items in a bag of 50 where X ~ B (50, 0.1)

mean = np = 50 x 0.1 = 5