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Probability

Question
CBSEENMA12033792

The random variable X has probability distribution P(X) of the following rm, where k is some number:
straight P left parenthesis straight X right parenthesis space equals space open curly brackets table row cell straight k comma space if space straight x space equals space 0 end cell row cell 2 straight k comma space if space straight x space equals space 1 end cell row cell table attributes columnalign left end attributes row cell 3 straight k comma space if space straight x space equals space 2 end cell row cell 0 comma space space otherwise end cell end table end cell end table close

(a)    Determine the value of k.
(b)    Find P(X < 2), P(X ≤ 2), P(X ≥ 2).

Solution

The probability distribution of X is

X

0

1

2

P(X)

k

2k

3k 


(a) We know that sum from blank to blank of straight P left parenthesis straight X right parenthesis space equals space 1
  therefore               straight k plus 2 straight k plus 3 straight k space equals space 1 space space space space space or space space space space 6 space straight k space equals 1 space space space rightwards double arrow space space space space straight k space equals space 1 over 6
(b)    straight P left parenthesis straight X less than 2 right parenthesis space equals space straight P left parenthesis straight X space equals space 0 right parenthesis space plus space straight P left parenthesis straight X space equals space 1 right parenthesis space equals space straight k space plus space space 2 space straight k space equals 1 over 6 plus 2 over 6 space equals 3 over 6 space equals 1 half
            straight P left parenthesis straight X space less or equal than space 2 right parenthesis space equals space straight P left parenthesis straight X space equals space 0 right parenthesis space plus space straight P left parenthesis straight X space equals space 1 right parenthesis space plus space straight P left parenthesis straight X space equals space 2 right parenthesis
                            equals straight k plus 2 straight k plus 3 straight k
equals space 1 over 6 plus 2 over 6 plus 3 over 6 space equals fraction numerator 1 plus 2 plus 3 over denominator 6 end fraction space equals 6 over 6 space equals space 1
straight P left parenthesis straight X space greater or equal than space 2 right parenthesis space equals space straight P left parenthesis straight X space equals 2 right parenthesis space equals space 3 over 6 space equals 1 half

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