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Stat Equation Sheet

Stat Equation Sheet - Chapter 2 Probability 1 If an...

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Chapter 2 – Probability 1. If an operation can be performed in n 1 ways, and for each of these a second operation can be performed in n 2 ways, and for each of the first two a third operation can be performed in n 3 ways, and so forth, then the sequence of k operations can be performed in n 1 n 2 …n k ways. 2. The number of permutations of n objects is n! 3. The number of permutations of n distinct objects taken r at a time is n P r = n ! ( n ! r )! . 4. The number of permutations of n objects arranged in a circle is (n-1)!. 5. The number of distinct permutations of n things of which n 1 are of one kind, n 2 of a second kind,..., n k of a kth kind is n ! n 1 ! n 2 ! ! n k ! . 6. The number of ways of partitioning a set of n objects into r cells with n 1 elements in the first cell, n 2 elements in the second, and so forth is n n 1 , n 2 ,... n r ! " # $ % & = n ! n 1 ! n 2 ! ! n r ! . 7. The number of combinations of n distinct objects taken r at a time is n r ! " # $ % & = n ! r !( n r )! . 8. If an experiment can result in any one of N different equally likely outcomes, and if exactly n of these outcomes correspond to event A, then the probability of event A is P ( A ) = n N . 9. The conditional probability of B, given A, denoted P ( B | A ) = P ( A ! B ) P ( A ) . 10. Two events A and B are independent if and only if P(B|A) = P(B) or P(A|B) = P(A). 12. Two events A and B are independent if and only if P ( A ! B ) = P ( A ) P ( B ) . 13. If the events B 1 , B 2, …, B k constitute a partition of the sample space S such that P(B i ) 0 for i = 1,2,…,k, then for any event A of S, P ( A ) = P ( B i ! A ) = P ( B i ) P ( A | B i ) i = 1 k " i = 1 k " . 14. (Bayes Rule) If the events B 1 , B 2 , …,B k constitute a partition of the sample space S such that P(B i ) 0 for i = 1,2,…,k, then for any event A in S such that P(A) 0, P ( B r | A ) = P ( B r ! A ) P ( B i ! A ) i = 1 k " = P ( B r ) P ( A | B r ) P ( B i ) P ( A | B i ) i = 1 k " . ||||| Chapter 3 – Random Variables and Probability Distributions 1. The set of ordered pairs (x, f(x)) is a probability function, probability mass function, or probability distribution of the discrete random variable X if, for each possible outcome x, a. f(x) 0, b. f ( x ) x ! = 1 , c. P(X=x) = f(x). 2. The cumulative distribution function F(x) of a discrete random variable X with probability distribution f(x) is F ( x ) = P ( X ! x ) = f ( t ) t ! x " for - < x< . 3. The function f(x) is a probability density function for the continuous random variable X, defined over the set of real numbers R, if a. f ( x ) ! 0 " x # R , b. f ( x ) dx = 1 !" " # , c. P ( a < X < b ) = f ( x ) dx a b ! . 4. The cumulative distribution function F(x) of a continuous random variable X with density function f(x) is F ( x ) = P X ! x ( ) = f ( t ) dt "# x $ for - < x< . 5. The function f(x,y) is a joint probability distribution or probability mass function of the discrete random variables X and Y if a. f(x,y) 0 for all (x,y), b. f ( x , y ) y ! = 1 x ! , c. P(X=x,Y=y) = f(x,y). 6. For any region A in the xy plane, P[(X,Y) A] = f ( x , y ) ! A ! . 7. The function f(x,y) is a joint density function of the continuous random variables X and Y if a. f(x,y) 0 for all (x,y), b. f ( x , y ) dxdy = 1 !" " # !" " # , c. P[(X,Y) A] = f ( x , y ) dxdy A !!
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