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 $r=1$, $2$, $3$, ...".