# 3 a ans the probability mass function can be written

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3. (a) (Ans) The probability mass function can be written by P ( Y i = y i ) = p · 1 { y i =1 } + q · 1 { y i =2 } + (1 - p - q ) · 1 { y i =3 } 2

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where 1 { y i = c } for c = 1 , 2 , 3 is an indecator fuctions defined as 1 { y i = c } = 1 for y i = c 0 otherwise . Let n j denote the number of data point at which y i = j for j = 1 , 2. I.e., n 1 is the number of data points for Y i = 1 while n 2 is the number of data points when Y i = 2. The number of data points for Y i = 3 automatically becomes n - n 1 - n 2 as Y i only takes three values 1 , 2 or 3 with positive probabilities. Given the probability mass function of Y i , the likelihood function is L n = p n 1 · q n 2 · (1 - p - q ) n - n 1 - n 2 and the log likelihood function is ln L n = n 1 ln p + n 2 ln q + ( n - n 1 - n 2 ) ln(1 - p - q ). (b) (Ans) Solving the following two equations simultaneously ln L n ∂p = n 1 p - n - n 1 - n 2 (1 - p - q ) = 0 ln L n ∂q = n 1 q - n - n 1 - n 2 (1 - p - q ) = 0, we obtain the MLE of p and q , ˆ p = n 1 n ˆ q = n 2 n . 3
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• Winter '08
• Stohs
• Normal Distribution, Probability theory, probability density function, Yi, Cumulative distribution function

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