### Pattern Recognition

When does a pattern start to exhibit structure? Can you crack the code used by the computer?

### Reaction Timer Timer

How can you time the reaction timer?

### Time to Evolve 2

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

# Can You Find ... Random Variable Edition

##### 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?

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

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