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## CY1B2 Statistics

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**CY1B2 Statistics**Aims: To introduce basic statistics. Outcomes: To understand some fundamental concepts in statistics, and be able to apply some probability distributions in applications. Statistics (5 lectures, 2 tutorials): Probability distributions, discrete and continuous distributions, Binomial distribution and Gaussian distributions and its applications.**Statistics**• The probability theory and statistics is a mathematical representation of random phenomena. • Random variable • The outcome of a random phenomenon may take a numerical value. When the outcomes of an event that produces random results are numerical, the numbers obtained are called random variables. • Random variable has probabilities associated with the various values of the variable. • Discrete random variable has a countable number of possible values.**Probability**• probability is always taken as a number lying between 0 and 1 and is denoted by p(x) • p(x) = 1 means that an event is certain to happen • p(x) = 0 means that an event is certain NOT to happen • so a toss of a coin could be represented as • P(heads) = 0.5 and P(tails) = 0.5 • or, more formally • p(x) = 0.5 x = heads, tails**Rules of probability**• there are some rules associated with probabilities when the events are not dependent on each other • P(A or B) = P(A + B) = P(A) + P(B) • P(A and B) = P(AB) = P(A)P(B) • for example, rolling a dice multiple times, or rolling two dice • if the events are not dependent but they are not mutually exclusive then • P(A + B) = P(A) + P(B) - P(AB) • how can this be extended to more than two possible events?**Discrete probability distributions**(i) Binomial distribution • An experiment which follows a binomial distribution will satisf the following requirements (think of repeatedly flipping a coin as you read these): • The experiment consists of n identical trials, where n is fixed in advance. • Each trial has two possible outcomes, S or F, which we denote ``success'' and ``failure'' and code as 1 and 0, respectively. • The trials are independent, so the outcome of one trial has no effect on the outcome of another. • The probability of success, p, is constant from one trial to another.**(i) Binomial distribution**The probability in a process with two outcomes; p for one outcome (e. g. success), q for another (e. g. failure) (q=1-p) The probability of r successes in n trials is given by the (r+1)th term of the binomial expansion of Where n! = 1 ×2×3×… ×n, factorial of n.**is the count of choosing r from n.**The coefficients of polynomial expansion can be listed as a pyramid as n=0; 1 n=1: 1 1 n=2; 1 2 1 n=3; 1 3 3 1 n=4; 1 4 6 4 1 n=5; 1 5 10 10 5 1 ………………………**Example: Toss a coin 4 times, p =q=0.5, n=4. Find the**probabilities of throwing 0,1,2,3,4 heads. A plot of p(r) versus r is called a probability distribution. The Figure on the left is the binomial probability distribution for this example (p=q=0.5) (Symmetrical)**Example: Toss a dice 4 times, Find the probabilities of**throwing 0,1,2,3,4 sixs. (p=1/6, q=5/6) The Figure on the left is the binomial probability distribution for this example (p=1/6, q= 5/6) (not symmetrical)**The mean value (average number of successes)**The mean square value can be shown to be The variance is a measure of the deviation from the mean, or the width of the distribution can be shown to be**The square root of the variance ,σ, in known as the**standard deviation. Example: Random sample of 900 people are asked “ have you heard of Cybernetics’’. Find mean, mean square and variance of the expected distribution, assuming that over the whole population, 1/3 would say “yes’’. 2/3 would say “no”.