Question

A) A certain area is hit by 6 Heat Waves a year on average. Find the probability that in a given year that area will be hit by fewer than 5 Heat Waves (5 is not included).

a) 0.2851

b) 0.4457

c) 0.1339

d) 0.1606

B) The probability of a successful rocket launch is 0.9. Test launches are conducted until three successful launches are achieved. Let X be the total number of launches needed. What kind of distribution does X follow?

C) What are the properties of a Bernoulli process?

Answer 1

Solution:-

A) (a) The probability that in a given year that area will be hit by fewer than 5 Heat Waves is 0.2851

x = 5

By applying poisons distribution:-

P(x < 5) = 0.2851

B) The X follows the Bernoulli distribution.

C)

The Bernoulli distribution is a special case of the binomial distribution where a single trial is conducted (so n would be 1 for such a binomial distribution).

In probability theory and statistics, the Bernoulli distribution, is the discrete probability distribution of a random variable which takes the value 1 with probability p and the value 0 with probability .

The properties of a Bernoulli process.

The Bernoulli distribution is a special case of the binomial distribution with

The kurtosis goes to infinity for high and low values of p, but for p = 1/2 the two-point distributions including the Bernoulli distribution have a lower excess kurtosis than any other probability distribution, namely −2.

The Bernoulli distributions for form an exponential family.

The maximum likelihood estimator of p based on a random sample is the sample mean.

The expected value of the random variable is given by E(x) = p and can be derived as follows

E(x) = 0*(1-p) + 1 * p = p

The variance of the Bernoulli variable is given by p*(1-p) and is given as follows:

Var(X) = p – p² = p*(1-p)