# Unit 8 - Statistics V3100018.001-18.006 UNIT 8 Discrete...

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Statistics – V3100018.001-18.006 UNIT 8 – Discrete random variables Giuseppe Arbia , Catholic University of the Sacred Hearth, Roma, Italy

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This week: Usual schedule. Homework 6 posted on thurdays, due back on Tuesday Next week: Homework 7 will be posted on Monday the 5th immediately after the lecture. Will not be corrected as usual. Not to be handed to the TA. Will be corrected in class on Wednesday the 7th in preparation to the MIDTERM EXAM of Friday 9th
The midterm is cumulative There will be one summary question on descriptive statistics and 5 questions on probability .

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A (Discrete) Random Variable A variable that assumes certain (discrete) values with a given probability A statistical Variables A variable that assumes certain (discrete) values with a given relative frequency.
Before the statistical experiment: a probability which could be known a priori. A random variable describes the population before the experiment. After the experiment: observed frequencies. The statistical variable describes a sample after observing it. Inferential procedures allow us to go from the Statistical variable to the random variable that generated it (with a certain degree of approximation that can be controlled).

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example Before tossing a coin we have a probability (that can be known or unknown) of obtaining a Head (a random variable) After tossing it several times we have the relative frequency of Heads in the sample (statistical varaible)
A probability distribution for a given random variable is the list of all possible outcomes of the variable and of the corresponding probability Experiment: Tossing 3 coins X = Total number of Heads in three trials X is a random variable Probability distribution of a discrete random variable P ( X = x ) is its probability distribution Capital X = the random variable Lower case x is the outcome of the Random variable In our case x = 0, 1, 2, 3 (number of Heads)

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( ) x P 3/8 2/8 1/8 0 1 2 3 x TTT HTT THT TTH HHT HTH THH HHH
A more formal definition: NOTICE: P(X) can be represented as: i) A table ii) A graph iii) An analytical formula A random variable is a function that associate to each elementary Event in the sample space S one and only one real number between 0 and 1

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To move from a sample space S to a numerical space that is easier to deal with. Logic: i) From S we move to P(X) ii) We forget about S iii) We work on the new space of probability which is easier to treat AIM
Example of no more than 1 Head in three draws. Sample space: HHH, THH,HTH,HHT,HTT,THT,TTH,TTT Event “no more than 1 Head” = E P(E) = ( ) ( ) ( ) 2 1 8 3 8 1 1 0 1 = + = + = P P X P

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X is a discrete random variable if it is defined on a discrete sample space S X can assume a finite number of outcomes or a countable number of infinite outcomes X is a Continuous random variable if it is defined on a continuous sample space S X can assume all infinite possible results within a given range ( t , T )
Definition: Given a discrete random variable X f x ( ) = P X = x ( ) Is called the probability function of X.

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