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Practice Problem: normalizing the probability mass function of a continuous random variable
A random variable X has the following probability density function:
$ f_X (x) = \left\{ \begin{array}{ll} k, & \text{ if } a\leq x \leq b,\\ 0, & \text{ else}, \end{array} \right. $
where k is a constant. Determine the cumulative distribution function of X.
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Answer 1
To find the CDF given the PDF, we must integrate:
$ F_X(x) = \int_{-\infty}^x \! f(t) \, \mathrm{d}t. $
However, before we integrate, we can setup our solution by knowing some properties of a cumulative distribution. Since we know that the cumulative distribution varies from 0 to 1 and that the provided pdf has a probability only in the range from a to b, we can infer that for any x less than a, the CDF will equal 0 and for any x greater than or equal to b, the CDF will equal 1 giving us this solution so far:
$ F_X (x) = \left\{ \begin{array}{ll} 0, & \text{ if } x < a,\\ ?, & \text{ if } a\leq x < b,\\ 1, & \text{ if } x \geq b, \end{array} \right. $
To solve for the ? between a and b, we perform the integral, however we do not need to integrate from negative infinity, we can simply integrate from the lower limit of a:
$ \int_{a}^x \! k \, \mathrm{d}t. $
Computing the integral we obtain:
$ kt\vert_a^x = k(x-a) $
Thus, the CDF is:
$ F_X (x) = \left\{ \begin{array}{ll} 0, & \text{ if } x < a,\\ k(x-a), & \text{ if } a\leq x < b,\\ 1, & \text{ if } x \geq b, \end{array} \right. $
Answer 2
Write it here
Answer 3
Write it here.
