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# Can You Find ... Random Variable Edition

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### Statistics - Maths of Real Life

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Age 16 to 18

Challenge Level

Can you find a discrete random variable $X$ which only takes integer values for which...

(a) ... the mean (expectation) is 2, the variance is 1, and $X$ takes only positive (integer) values? Can you find another? And a very different example?

(b) ... the mean is 2, the variance is 1, and $X$ takes negative or zero values in addition to positive values? Can you find another? And a very different example?

*The next two parts are significantly more challenging. You might find it helpful to look at A Swiss Sum if you are stuck.*

(c) ... $X$ takes only positive values, but its mean is infinite? Can you find another?

(d) ... the mean is 0 and the variance is as large as possible?

*This resource is part of the collection Statistics - Maths of Real Life*

(a) ... the mean (expectation) is 2, the variance is 1, and $X$ takes only positive (integer) values? Can you find another? And a very different example?

(b) ... the mean is 2, the variance is 1, and $X$ takes negative or zero values in addition to positive values? Can you find another? And a very different example?

(c) ... $X$ takes only positive values, but its mean is infinite? Can you find another?

(d) ... the mean is 0 and the variance is as large as possible?

You could give your answers in the form of a probability distribution table for $X$, or as a rule such as "$\mathrm{P}(X=r)=\cdots$ for $r=1$, $2$, $3$, ...".

How is the length of time between the birth of an animal and the birth of its great great ... great grandparent distributed?

This pilot collection of resources is designed to introduce key statistical ideas and help students to deepen their understanding.