Statistical Sciences 1024A/B Lecture Notes - Lecture 12: Conditional Probability

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silverprairie-dog97

20 Apr 2016

School

Western University

Department

Statistical Sciences

Course

Statistical Sciences 1024A/B

Professor

Lori Murray

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Document Summary

Two events are disjoint if they have no outcomes in common and can never happen together. Example: if you flip a coin once, it may turn out heads or tails. However, you cannot obtain both heads and tails on the same flip. P(a or b) = p(a) + p(b) - p(a and b) Outcomes common to both are counted twice when we add probabilities, so we must subtract this probability once. There are 4 aces in the pack and 13 hearts. However, one card is both an ace and a heart. P(ace or heart) = p(ace) + p(heart) p(ace and heart) P(domestic or light truck) = p(domestic) + p(light truck) p(domestic light truck) The probability of one event occurring, given that another event has occurred is called a conditional probability. The conditional probability of b given a is denoted by p(b|a) | = given the information that o: the proportion of all occurrences of a for which b also occurs.

Related Questions

You roll two fair dice, one green and one red.

(a) Are the outcomes on the dice independent? Yes No

(b) Find P(5 on green die und 3 on red die). (Enter your answer as a fraction.)

(d) Find P((5 on green die and 3 on red die) or (3 on green die and 5 on red die)). (Enter your answer as a fraction.)

emeraldsea-lion69

You are given a fair die, i.e. the outcomes of any throw are contained in the sample space Let the event A be that an even number is thrown and the event B that a multiple of three is thrown. Now consider the two random variables X and Y. X takes on a value of 1 if the event A has occurred and the value 0 if A has not occurred. The random variable Y takes on the value of 1 if the event B has occurred and the value zero if B has not occurred.

(a) Derive the joint pdf of X and Y. Provide this in the form of a table

Y=0 Y=1

X=0
X=1

b) What is the marginal probability distribution of Y?
(c) What is the conditional probability distribution of X, given that Y = 1?
(d) What is E (X|Y = 1)?
(e) What is Cov (X, Y )?
(f) Are X and Y independent random variables?