15_Exam2_review_Spring_2012

This number is called the numerical value or the

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Unformatted text preview: group has ni items (where n1 + n2 + … + nr = n)? ⎛ ⎞ n n! Answer: ≡⎜ ⎟ = multinomial coefficients. n1 !n2 !… nr ! ⎝ n1 , n2 ,…, nr ⎠ The reason for this name is that ( x1 + x2 + … + xr ) n ⎛ ⎞n n n = ∑ ⎜ n1, n2 ,…, nr ⎟ x1 1 x2 2 ⋅… ⋅ xrnr n1 + n2 +…+ nr = n ⎝ ⎠ •  Example: if r = 2, the answer is Ilya Pollak Random Variables: DefiniAon •  A random variable is an assignment of a value (number) to every outcome in the sample space. This number is called the numerical value or the experimental outcome of the random variable. •  In other words, a random variable is a funcAon from the sample space to the set of real numbers. Ω R Ilya Pollak Probability Mass FuncAon (PMF) •  Aka “probability law,” “probability distribuAon” •  PMF of a random variable X is the following funcAon: pX ( x ) = P( X = x ) Ilya Pollak Expected value of X E[ X ] = ∑ xp X ( x ) x E [ X ] is also called the mean of X Ilya Pollak Linearity of expectaAon Ilya Pollak Variance and Standard DeviaAon of X ⎡( X − E[ X ])2 ⎤ = E[ X 2 ] − ( E[ X ])2 var( X ) = E ⎣ ⎦ Standard deviation of X : σ X = var( X ) Ilya Pollak FuncAons of Random Variables •  If X is a random variable, then Y=g(X) is also a random variable. Y Ω g X x y Ilya Pollak Joint PMF pX 1 ,X 2 ,…,X n ( x1, x 2 ,…, x n ) = P( X1 = x1, X 2 = x 2 ,…, X n = x n ) The marginal PMF for each X i can be obtained from the joint PMF by...
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