Hypergeometric distribution: male chickens
It is notoriously difficult to determine the sex of a chicken before it matures. Suppose that a randomly chosen chick from a pen of
chicks is male with probability . Suppose you draw chicks from the pen. Do the odds that of them are male depend upon the value of ?
Components of a car
A certain car includes
components, and it needs at least half of its components to function in order to drive. Each component functions with probability , independently of the others. For what values of is a car with more likely to drive than a car with ? Start by defining a random variable that measures the number of functioning components.
Construct random variable from PMF
The probability space
is the set of infinite strings of or , for example . There is a probability measure determined by the interpretation of or as the outcome of a fair coin flip. For a subset containing all strings with specified (given) values of the first letters, then . E.g. if is everything starting with , then . Construct a random variable
on such that the PMF of is given by:
Fair coin,
flips, Repeat the previous example with a biased coin having
.
Random point in
, PDF of and A random point is chosen in the unit square
. Outcomes are points
in this square. Events are regions in the square. The probability of a region is the area of . Define the random variable
by . This is just the -coordinate of the random point. Then the random variable is given by . This is just the squared -coordinate of the random point.
- (a) Describe the PDF of
. - (b) Describe the PDF of
.