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Mathematical Methods Outcome 6- Probability T wo -state Ma rko v C ha ins: the result of each trial is conditional on the result of the previous trial C ontinuous prob abil ity: it can take any value in a given interval The general structure of a transition matrix: S te ady state o f a Marko v chain: The properties of the probability density function: 1. f(x)≥0 for all x ! 2. ∫!! 𝑓(𝑥)𝑑𝑥 = 1 Geo metric Dis trib utio n: It is different to binomial and markov chain. Typically a question will ask: e.g. What is the probability that you have x failures before success: Tip: 1. f(x) does not have to be smooth curve. They can be other shapes too! Ex pected value o f a c ontinuous prob abili ty dens ity func tio n: N ormal Di stri bution: it may take any value within a given range, and no distinct, countable values. Variance It has the following properties: 1. Symmetrical 2. Bell shaped 3. Mean=mode=median S ta nd ard isa ti on and the 6 8-9 5-9 9.7 % rule P erce ntile and the media n The median for continuous random variable is such that: 99.7% of the values lie within three standard deviation of the means We can solve this on the calculator (CAS): 95% of the values lie within two standard deviation of the mean The interquartile range of X is: IQR= b-a 68% of the values lie within one standard deviation of the mean Calculator tips: We can determine normal probabilities using this To standardise, we use Z= !!! !