Unit 7: Standard Distribution - Subjective Questions

A random variable follows a binomial distribution when it counts the number of successes in independent Bernoulli trials.

Assumptions:

• The number of trials is fixed.
• Each trial has two possible outcomes: success or failure.
• Trials are independent.
• The probability of success is constant for every trial.

If , its probability mass function is

Its principal measures are:

• Mean:
• Variance:
• Standard deviation:

The notation used is .

For , let

Then

Therefore, the recurrence relation is

Probabilities increase while . This gives

which simplifies to . Hence:

• If is not an integer, the unique mode is .
• If is an integer , the two modes are and .

Let denote the number of heads. Then

(a) Exactly four heads:

Therefore,

(b) At least one head:

Using the complementary event,

Thus,

Hence, the required probabilities are approximately and .

Let be the number of successes in independent Bernoulli trials. Define

Then

For each indicator, and .

By linearity of expectation,

Since the trials are independent, the indicators are independent, so all covariance terms are zero. Therefore,

Consequently:

• Mean:
• Variance:
• Standard deviation:

For a binomial distribution,

Given and ,

so . Thus,

For observations, the expected frequency is .

• :
• :
• :
• :
• :
• :

These expected frequencies total .

A random variable has a Poisson distribution with parameter if

It is denoted by .

Major properties:

• .
• .
• The standard deviation is .
• Its moment-generating function is
  
• Its skewness is , so it becomes more symmetric as increases.
• Independent Poisson variables are additive.

Applications: It models counts of relatively rare, independent events occurring in a fixed interval of time, distance, area, or volume, such as telephone calls, defects, accidents, or customer arrivals.

Let and suppose

where remains finite. Put . Then

Rewrite this as

For fixed , as :

and

Therefore,

which is the probability mass function of .

For , write

The ratio of successive probabilities is

Hence, the recurrence relation is

starting with .

The probabilities increase while and decrease after this ratio becomes less than . Therefore:

• If is not an integer, the unique mode is .
• If is a positive integer, the two modes are and .

This recurrence is also useful for calculating probabilities without repeatedly evaluating factorials.

Let be the number of calls in one minute. Then

(a) Exactly two calls:

(b) More than two calls:

Now,

Therefore,

Thus, the required probabilities are approximately and .

Suppose are independent and

The moment-generating function of is

Let

Because the variables are independent, the MGF of their sum is the product of their MGFs:

Therefore,

This is the MGF of a Poisson distribution with parameter

Hence,

Consider independent Bernoulli trials having success probability and failure probability . Let be the number of failures occurring before the th success. Then has a negative binomial distribution.

Its probability mass function is

It is denoted by under this parameterization.

Important properties:

• Mean:
• Variance:
• MGF:
  
  provided .
• The variance exceeds the mean when , making the distribution useful for overdispersed count data.
• When , it reduces to the geometric distribution.

Let be the number of failures before the th success. Write

where is the number of failures after the th success and before the th success. Each has the geometric probability mass function

For this geometric distribution,

The variables are independent. Therefore,

and

Thus, for the number of failures before the th success:

• Mean:
• Variance:

For , where counts failures before the th success,

The ratio of successive probabilities is

Thus, the recurrence relation is

with .

For , define

• If is not an integer, the unique mode is .
• If is a positive integer, the two adjacent modes are and .
• For , the distribution is geometric and its mode is .

The recurrence relation provides an efficient way to generate successive probabilities.

Let be the number of failures before the third success. Then

Its probability mass function is

(a) Exactly two failures:

Thus,

(b) At most two failures:

Therefore,

Hence, the required probabilities are and .

Binomial distribution:

• Counts successes in a fixed number of independent trials.
• Parameters are and .
• Support is .
• Mean and variance are and .

Poisson distribution:

• Counts events in a fixed interval when events occur independently at a constant average rate.
• Its parameter is .
• Support is .
• Mean and variance are both .

Negative binomial distribution:

• Counts failures before a fixed number of successes.
• Parameters are and .
• Support is .
• Mean and variance are and .

The binomial fixes the number of trials, the negative binomial fixes the number of successes, and the Poisson models event counts over an exposure interval.

A continuous random variable has a normal distribution with mean and variance if its probability density function is

It is written as .

Properties:

• The curve is bell-shaped, continuous, and symmetric about .
• Mean, median, and mode are all equal to .
• The total area under the curve is .
• Half of the area lies on each side of .
• The tails approach but never meet the horizontal axis.
• The points of inflection are and .
• Approximately , , and of observations lie within , , and standard deviations of the mean, respectively.
• Its skewness is and kurtosis is .

Let . Standardize using

(a) Scores between and :

For ,

For ,

Thus,

Using standard normal values,

(b) Scores above :

Therefore, approximately score between and , while score above .

Let . Its MGF is

Combining exponents and completing the square gives

The remaining integral is the integral of a normal density and equals . Hence,

Differentiating,

so . Also,

Therefore,

Thus, the parameters and are respectively the mean and variance.

For the binomial distribution,

and

so . Since both and are sufficiently large, a normal approximation is appropriate.

Using the continuity correction,

where .

The corresponding standard scores are

and

Therefore,

Thus, the required probability is approximately .

If , its standardized form is

Then , called the standard normal distribution. Standardization expresses each observation as the number of standard deviations it lies from the mean.

For a symmetric central interval containing probability , choose such that

Then

Transforming back to gives

For example, , so the central interval is

The th percentile of is

where .