By R. Meester

In this advent to likelihood idea, we deviate from the path often taken. we don't take the axioms of likelihood as our start line, yet re-discover those alongside the way in which. First, we talk about discrete likelihood, with purely chance mass capabilities on countable areas at our disposal. inside this framework, we will already speak about random stroll, vulnerable legislation of huge numbers and a primary vital restrict theorem. After that, we commonly deal with non-stop chance, in complete rigour, utilizing merely first 12 months calculus. Then we speak about infinitely many repetitions, together with robust legislation of huge numbers and branching strategies. After that, we introduce susceptible convergence and end up the important restrict theorem. eventually we encourage why yet another research will require degree idea, this being the proper motivation to review degree concept. the speculation is illustrated with many unique and incredible examples.

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**Example text**

The results (diﬀerentiated between men and women) were as follows. women succes failure drug I 200 1800 drug II 10 190 men succes failure drug I 19 1 drug II 1000 1000 We are now faced with the question which drug is better. Here are two possible answers: (1) Drug I was given to 2020 people, of whom 219 were cured. Drug II was given to 2200 people, of whom 1010 were cured. Therefore, drug II is much better, (2) Amongst women, the succes rate of drug I is 1/10, and for drug II the succes rate is 1/20.

2. Let X be a random variable with P (X = 0) = 1/2, and P (X = 1) = 1/2. Then E(X) = 1/2. This shows that the expectation of a random variable X need not be a value which can be taken by X. 3. Suppose that X takes values -1, 1 and 2, with equal probability, and consider the random variable Y = X 2 . Clearly, Y takes values 1 and 4 with probability 23 and 13 respectively. Therefore E(Y ) = 1 · 23 + 4 · 13 = 2. 4 (Poisson distribution). Recall that a random variable X is said to have a Poisson distribution with parameter λ > 0 if P (X = k) = λk −λ e , k!

If the ball is red, we draw a second ball from another urn containing 4 red and 1 blue ball. If the ﬁrst ball is blue, we draw a second ball from an urn with just 4 blue balls. (a) What is the conditional probability that the second ball is red, given that the ﬁrst ball is red? (b) What is the probability that both balls are red? (c) What is the probability that the second ball is blue? 19. Suppose that we send a message using some coding so that only 0’s and 1’s are sent. On average, the ratio between the number of 0’s and 1’s that are sent is 3/4.