Please have a go at the problems before the workshop so that in the workshop you can focus on those problems which you had trouble with.
The brave knight is lost in the Tulgey Wood. Three paths lead on from where he stands at the Tumtum tree, clutching his vorpal sword. Unbeknownst to him, all three paths lead out of the wood; however, just off the left path lies the Jubjub bird, sleeping. Two times in three a knight walking this path would pass the bird without waking it, but if awoken the Jubjub bird would carry him up to its nest of hungry young. Along the right path lurks the frumious Bandersnatch against which the knight has no hope of prevailing. The middle path leads to the Jabberwock; if the knight encounters the Jabberwock then he might fall victim to its jaws that bite, its claws that snatch or its beguiling eyes of flame; else he will defeat it with a snicker-snack of his vorpal sword; each of the four possibilities is equally likely. The knight is as likely as not to forge straight ahead down the centre path; the right and left paths appeal equally to him.
What is the probability that the knight makes it out of the tulgey wood alive?
The brave knight does not emerge from the wood; what is the probability that he encountered the Jabberwock?
Hint: It may be helpful to define: ‘A’=‘Alive’, ‘D’=‘Dead’, L=‘left path’, R=‘right’ and C=‘centre’.
An ectopic pregnancy is twice as likely to develop when a pregnant woman is a smoker as it is when she is a non smoker. If 32% of women of childbearing age are smokers, what is the probability that a woman having an ectopic pregnancy is a smoker? You may find it helpful to define the following shorthand: , .
A discrete random variable, , has a cdf of
What is ?
What is ?
What is the pmf of ?
What is the median of ?
Obtain the cdf of a random variable with the following pmf.
Hint Study Figure 2.1 in your notes. Your function should work for any real number . First calculate at the integers, being careful to distinguish the index of summation from the upper limit of summation. Secondly extend from the integers to the real line.
Let the cdf of a continuous random variable be
Find the pdf of .
The random variable has pdf
Find .
Find .
The random variable has probability density function
Find the constant, , and the cumulative distribution function of and, hence or otherwise, evaluate .
Find the median, and of .