Mathematical statistics and data analysis

Please make sure to include the thought and comments how to solve these
questions so I can understand them, thanks.

Two boys play basketball in the following way. They make turns shooting
and stop when a basket is made. Player A goes first and has and has
probability p1 of making a basket on any throw. Player B, who shoots
second, has probability p2 of making a a basket. The outcomes of
successive trials are assumed to be independent.

Find the frequency function for the total number of attemps.

What is the probability that player A wins?

If X is a geometric random variable, show that

P ( X > n + k – 1│X > n – 1 ) = P ( X > k )

In light of the construction of a geometric distribution from a sequence
of independent Bernoulli trials, how can this be independent so that it
is “obvious”?

Phone calls are received at a certain residence as a Piosson process
with parameter (=2 per hour.

If Diane takes a 10 min shower, what is the probability that the phone
rings during that time?

How long can her shower be if she wishes the probability of receiving no
phone calls to be at most 0.5 ?

If U is a uniform random variable on [ 0 , 1 ], what is the distribution
of random variable X = [ nU ], where [ t ] denotes the greatest integer
less than or equal to “t” ?

Suppose that X has the density function f(x) = cx2 for 0 ( x ( 1 and f
(x) = 0 other wise.

Find c.

Find the cumulative distribution function (cdf).

What is P ( 0.1 ( X ( 0.5 ) ?

Suppose that the lifetime of an electric component follows an
exponential distribution with ( = 0.1 .

Find the probability that the lifetime is less than 10.

Find the probability that the lifetime is between 5 and 15.

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What proportion of the population is over 6ft tall?

What is the distribution of heights if they are expressed in
centimeters? In meters?

8. If the radius of a circle is an exponential random variable, find
the density function area.