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I saw other similar posts where they do the integration from math stackexchange, but confused about this one since it is given in p. The following is my problem.

I was looking to find the mean from the following cumulative distribution function.

The function F(x) is

0 when z<o

p when o $\leq$ z<2

1 when z$\geq$ 2

The answer given in the solution is 2(1-p) but I couldn't follow the steps. I saw other similar posts where they do the integration from math stackexchange, but confused about this one since it is given in p.

appreciate your support. Thanks!

Alexa Thomas
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  • For a non-negative random variable, $\mathbb E[X]=\int\limits_{x=0}^\infty (1-F(x)), dx$, which is quite easy here – Henry Apr 10 '21 at 16:50

1 Answers1

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Your rv is discrete taking only the values $\{0;2\}$ with probability $\{p;(1-p)\}$, respectively, thus the expectation is

$$\mathbb{E}(X)=0\times p+2\times(1-p)=2(1-p)$$

To understand that your rv is discrete do a drawing of your CDF and observe that it has two "jumps" (two discontinuity points) in $x=0$ and $x=2$ and the sum of the two jumps is 1

The expectation is the area above the CDF: the area of a rectangle with base $2$ and height $(1-p)$

tommik
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