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Probability

Question
CBSEENMA12033824

Let X denote the sum of the numbers obtained when two fair dice are rolled. Find the variance and standard deviation of X.

Solution
Let X denote the sum of the numbers obtained on the two dice. X can take values 2, 3, ....., 12
  straight P left parenthesis straight X space equals space 2 right parenthesis space equals space straight P thin space left curly bracket left parenthesis 1 comma space 1 right parenthesis right curly bracket space equals space 1 over 36
straight P left parenthesis straight X space equals space 3 right parenthesis space equals space straight P left curly bracket left parenthesis 1 comma space 2 right parenthesis comma space left parenthesis 2 comma space 1 right parenthesis right curly bracket space equals space 2 over 36
straight P left parenthesis straight X space equals space 4 right parenthesis space equals space straight P left curly bracket left parenthesis 1 comma space 3 right parenthesis comma space left parenthesis 2 comma space 2 right parenthesis comma space left parenthesis 3 comma space 1 right parenthesis right curly bracket space equals space 3 over 36
straight P left parenthesis straight X space equals space 5 right parenthesis space equals space straight P left curly bracket left parenthesis 1 comma space 4 right parenthesis comma space left parenthesis 2 comma space 3 right parenthesis comma space left parenthesis 3 comma space 2 right parenthesis comma space left parenthesis 4 comma space 1 right parenthesis right curly bracket space equals space 4 over 36
straight P left parenthesis straight X space equals space 6 right parenthesis space equals space straight P left curly bracket left parenthesis 1 comma space 5 right parenthesis comma space left parenthesis 2 comma space 4 right parenthesis comma space left parenthesis 3 comma space 3 right parenthesis comma space left parenthesis 4 comma space 2 right parenthesis comma space left parenthesis 5 comma space 1 right parenthesis right curly bracket space equals space 5 over 36
straight P left parenthesis straight X space equals space 7 right parenthesis space equals space straight P left curly bracket left parenthesis 1 comma space 6 right parenthesis comma space left parenthesis 2 comma space 5 right parenthesis comma space left parenthesis 3 comma space 4 right parenthesis comma space left parenthesis 4 comma space 3 right parenthesis comma space left parenthesis 5 comma space 2 right parenthesis comma space left parenthesis 6 comma space 1 right parenthesis right curly bracket space equals space 6 over 36
straight P left parenthesis straight X space equals space 8 right parenthesis space equals space straight P left curly bracket left parenthesis 2 comma space 6 right parenthesis comma space left parenthesis 3 comma space 5 right parenthesis comma space left parenthesis 4 comma space 4 right parenthesis comma space left parenthesis 5 comma space 3 right parenthesis comma space left parenthesis 6 comma space 2 right parenthesis right curly bracket space equals space 5 over 36
straight P left parenthesis straight X space equals space 9 right parenthesis space equals space straight P left curly bracket left parenthesis 3 comma space 6 right parenthesis comma space left parenthesis 4 comma space 5 right parenthesis comma space left parenthesis 5 comma space 4 right parenthesis comma space left parenthesis 6 comma space 3 right parenthesis right curly bracket space equals space 4 over 36
straight P left parenthesis straight X space equals space 10 right parenthesis space equals space straight P left curly bracket left parenthesis 4 comma space 6 right parenthesis comma space left parenthesis 5 comma space 5 right parenthesis comma space left parenthesis 6 comma space 4 right parenthesis right curly bracket space equals space 3 over 36
straight P left parenthesis straight X space equals space 11 right parenthesis space equals space straight P left curly bracket left parenthesis 5 comma space 6 right parenthesis comma space left parenthesis 6 comma space 5 right parenthesis right curly bracket space equals space 2 over 36
straight P left parenthesis straight X space equals space 12 right parenthesis space equals space straight P left curly bracket left parenthesis 6 comma space 6 right parenthesis right curly bracket space equals space 1 over 36
Probability distribution of X is

Mean X = sum from blank to blank of straight X thin space straight P left parenthesis straight X right parenthesis
     equals space fraction numerator 2 space cross times 1 plus 3 cross times 2 plus 4 cross times 3 plus 5 cross times 4 space plus space 6 space cross times 5 space plus space 7 space cross times 6 space plus space 8 space cross times 5 space plus space 9 cross times 4 space plus space 10 space cross times 3 space plus 11 space cross times 2 space plus 12 space cross times space 1 over denominator 36 end fraction
space equals 252 over 36 space equals space 7
Variance straight X space equals space sum from blank to blank of straight X squared space straight P left parenthesis straight X right parenthesis space minus space left parenthesis mean right parenthesis squared
                   fraction numerator left parenthesis 2 squared cross times 1 space plus space 3 squared space cross times 2 space plus space 4 squared cross times 3 space plus space 5 squared space cross times space 4 space plus space 6 squared space cross times space 5 space plus space 7 squared space cross times 6 space plus space 8 space squared space cross times 5 plus 9 squared space cross times 4 space plus space 10 squared space cross times 3 space plus space 11 squared cross times 2 space plus space 12 squared space cross times 1 right parenthesis over denominator 36 end fraction minus 7 squared
 = fraction numerator 4 plus 18 plus 48 plus 100 plus 180 plus 294 plus 320 plus 324 plus 300 plus 242 plus 144 over denominator 36 end fraction minus 49
 = 1974 over 36 minus 49 space equals space fraction numerator 1974 minus 1764 over denominator 36 end fraction space equals 210 over 36 space equals space 35 over 6.
therefore space space straight S. straight D. space space equals space square root of Variance space equals square root of 35 over 6 end root space equals space 1.4

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