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Chapter 2

Q. 2.6

Find the probability density function for the random variable X of Example 2.2, for which

F_{Y}(u)=0 For u \lt 0
F_{Y}(u)=0.05(u+4) For 0 \leq u \lt 10
F_{X}(u)=1 For u \geq 10

 

Step-by-Step

Verified Solution

Differentiating the cumulative distribution function, the probability density function is found to be

p_{X}(u)=0.1 For 0 \leq u \lt 10
p_{X}(u)=0 otherwise

or, equivalently,

p_{X}(u)=0.1U(u)U(10-u)            or              p_{X}(u)=0.1[U(u)-U(u-10)]

Note that the final two forms differ only in their values at the single point u=10. Such a finite difference in a probability density function at a single point (or a finite number of points) can be considered trivial, because it has no effect on the cumulative distribution function, or any other integral of the density function.