Stats 241 Assignment 5 – Solutions

  1. A random variable, X, has probability density function

a)Find a.

Solution:

Thus

b)Find P(X  4).

  1. Suppose X has the distribution function

a)Find a.

Solution: Since X is a continuous random variable (It has a density function), Then F(x) is continuous. This will be true if

Thus

b)Find P(X  1)

Solution:

c)Find f(x) the density function of X.

Solution:

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Stats 241 Assignment 5 – Solutions

  1. In preparation for a long weekend, a hospital in an National park is purchasing antidotes for rattle snake bites. Past experience has shown that the number of rattle snake bites occurring during this long weekend has a Poisson distribution with mean = 0.80. How many antidotes should the hospital have on hand so that there is at least a 99.99% chance that an antidote will be available to all individuals who suffer a rattle snake bite. How should this be altered if the mean number of rattlesnake bites could be as high as =3.1.

Solution: Let X = # of snake bites during the long weekend.

We need to find c such that

Here is a table of x, p(x) and F(x):

  1. The expected number of suicides in a large metropolitan city in a month has is known to be = 4.5. The number of suicides is known and to have a Poisson distribution.

a)Compute the probability of at least 8 suicides in a month

Solution:

=0.08659

b)Compute the probability of at most 10 suicides in a month, if it is known that there were at least 3 suicides. (Assume that the number of suicides for a month in the city follows a Poisson distribution)

.Solution:

  1. Suppose that a random variable X, has probability density function.

determine E(X) and E(X2).

Solution

  1. Suppose that X is a Poisson random variable with parameter .

Find  if P(X = 2) = P(X = 3).

Solution

and suppose that p(2) = p(3)

Then or .

  1. Show, if X is a Poisson random variable with parameter  where  is an integer, that some 2 consecutive values of X have equal probabilities.

Solution

if . Thus p(x) = p(x + 1) if x = – 1.

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Stats 241 Assignment 5 – Solutions

  1. Let X be Poisson with parameter .

a)Find a recursion for P(X = x + 1) in terms of P(X = x).

Solution

Thus

Hence the recursion formula is:

b)Use the recursion in part a) to find  and 2.

Solution

Note: and

Let then

Also

And

Hence and

  1. Ten people are wearing tags numbered 1, 2,..., 10. Three people are chosen at random. Let X = the smallest badge number among the three. Find the probability distribution of X.

Solution

Note the possible values of X are x = 1, 2, 3, 4, 5, 6, 7 and 8.

In order for x to be the smallest number it has to be chosen and the remaining 2 numbers have to be chosen from the 10 – x numbers larger than x.

The number of ways of choosing 3 numbers from 10 is:

The number of ways of choosing 3 numbers from 10 so that x is the smallest is:

Thus for x = 1 to 8.

x / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8
p(x) / / / / / / / /
  1. Let X be a random variable with probability density function

a)Find k.

Solution:

, hence

b)Calculate E(X).

Solution:

c)Find the cumulative distribution function, F(x).

Solution:

  1. The percentage X of antiknock additive in a particular gasoline is a random variable with probability density function

a)Find k.

Solution:

hence

b)Evaluate P[X < E(X)].

Solution:

c)Find F(x).

Solution:

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