document.write( "Question 1165673: Five-digit codes are selected at random from the set {0, 1, 2, ..., 9} with replacement. If the random variable X denotes the number of zeros in randomly chosen codes, then what are the space and the probability density function of X? \n" ); document.write( "
Algebra.Com's Answer #852027 by ikleyn(52781)\"\" \"About 
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\n" ); document.write( "Five-digit codes are selected at random from the set {0, 1, 2, ..., 9} with replacement.
\n" ); document.write( "If the random variable X denotes the number of zeros in randomly chosen codes, then what are
\n" ); document.write( "the space and the probability density function of X?
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document.write( "The words \"with replacement\" mean, in this context, that in every of 5 positions, from 1 to 5,\r\n" );
document.write( "from left to right, any of 10 digits can appear independently of digits in other positions, \r\n" );
document.write( "with the probability  \"1%2F10\" = 0.1.\r\n" );
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document.write( "For X, we have 6 possible values, X = 0, 1, 2, 3, 4, 5.\r\n" );
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document.write( "The probability density function of X is\r\n" );
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document.write( "    P(X=0) = \"%289%2F10%29%5E5\" = \"0.9%5E5\" = 0.59049,\r\n" );
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document.write( "    P(X=1) = \"C%285%2C1%29%2A0.1%5E1%2A0.9%5E4\" = \"0.5%2A0.9%5E4\" = 0.32805,\r\n" );
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document.write( "    P(X=2) = \"C%285%2C2%29%2A0.1%5E2%2A0.9%5E3\" = \"10%2A0.01%2A0.9%5E3\" = \"0.1%2A0.9%5E3\" = 0.0729,\r\n" );
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document.write( "    P(X=3) = \"C%285%2C3%29%2A0.1%5E3%2A0.9%5E2\" = \"10%2A0.1%5E3%2A0.9%5E2\" = \"0.01%2A0.9%5E2\" = 0.0081,\r\n" );
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document.write( "    P(X=4) = \"C%285%2C4%29%2A0.1%5E4%2A0.9%5E1\" = \"5%2A0.1%5E4%2A0.9\" = \"0.0005%2A0.9\" = 0.00045,\r\n" );
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document.write( "    P(X=5) = \"C%285%2C5%29%2A0.1%5E5%2A0.9%5E0\" = \"0.1%5E5\" = 0.00001.\r\n" );
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document.write( "CHECK.  0.59049 + 0.32805 + 0.0729 + 0.0081 + 0.00045 + 0.00001 = 1.   (! correct !)\r\n" );
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