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In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function that describes the relative likelihood for this random variable to take on a given value.
Probability density function is defined by following formula:
${P(a le X le b) = int_a^b f(x) d_x}$
Where −
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${[a,b]}$ = Interval in which x lies.
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${P(a le X le b)}$ = probability that some value x lies within this interval.
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${d_x}$ = b-a
Example
Problem Statement:
During the day, a clock at random stops once at any time. If x be the time when it stops and the PDF for x is given by:
${f(x) =
begin{cases}
1/24, & text{for $ 0 le x le 240 $} \
0, & text{otherwise}
end{cases} }$
begin{cases}
1/24, & text{for $ 0 le x le 240 $} \
0, & text{otherwise}
end{cases} }$
Calculate the probability that clock stops between 2 pm and 2:45 pm.
Solution:
We have found the value of the following:
${P(14 le X le 14.45) = int_{14}^{14.45} f(x) d_x \[7pt]
= frac{1}{24} (14.45 – 14) \[7pt]
= frac{1}{24}(0.45) \[7pt]
= 0.01875 }$
= frac{1}{24} (14.45 – 14) \[7pt]
= frac{1}{24}(0.45) \[7pt]
= 0.01875 }$
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