SOLUTION: 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 ar

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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?
Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
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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?
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The words "with replacement" mean, in this context, that in every of 5 positions, from 1 to 5,
from left to right, any of 10 digits can appear independently of digits in other positions, 
with the probability  1%2F10 = 0.1.


For X, we have 6 possible values, X = 0, 1, 2, 3, 4, 5.


The probability density function of X is

    P(X=0) = %289%2F10%29%5E5 = 0.9%5E5 = 0.59049,

    P(X=1) = C%285%2C1%29%2A0.1%5E1%2A0.9%5E4 = 0.5%2A0.9%5E4 = 0.32805,

    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,

    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,

    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,

    P(X=5) = C%285%2C5%29%2A0.1%5E5%2A0.9%5E0 = 0.1%5E5 = 0.00001.


CHECK.  0.59049 + 0.32805 + 0.0729 + 0.0081 + 0.00045 + 0.00001 = 1.   (! correct !)

Solved.