Topic 3&4 - Discrete Random Variables

Question 1

What is a random variable

Answer 1

any chance situation
where outcomes are always numerical

Question 2

What is a discrete random variable

Answer 2

can take only some values in a range, e.g., die score is
one of 1, 2, 3, 4, 5 or 6 only

Question 3

What is continuous random variable

Answer 3

Continuous Random Variables: can take
any value within a range, e.g., height,
weight, income

Question 4

What is expected value or expectation?

Answer 4

The mean of a random variable

Question 5

What is binomial trial?

Answer 5

• Binomial trial (or Bernoulli trial) is a random experiment with exactly
two possible outcomes, “success” and “failure”, in which the
probability of success is the same every time the experiment is
conducted.
• “Success”: indicates the outcome of interest
• “Failure”: usually the complement of the “success” →Not “success”

Question 6

What is binomial experiment with n trials?

Answer 6

• The experiment is repeated a fixed number of times (n times).
• Each trial has only two possible outcomes, “success” and “failure”.
The possible outcomes are exactly the same for each trial.
• The probability of success (𝒑) remains the same for each trial. We
use 𝑝for the probability of success (on each trial) and 𝑞=1−𝑝 for the
probability of failure.
• The trials are independent (the outcome of previous trials has no
influence on the outcome of the next trial).

Question 7

What is Poisson distribution ?

Answer 7

• Poisson distribution, in particular, is used in queueing theory to model
the number of people who join a queue during a particular time
interval. There are two broad sets of circumstances for which a
Poisson distribution is appropriate:
• to model the number of occurrences of a particular rare event in a time
interval
• as an approximation to the binomial distribution

Question 8

What is the Poisson distribution used for?

Answer 8

• The Poisson distribution is often appropriate for modelling the
number of times a particular rare event happens in a time interval.
For instance,
• the number of road accidents on a particular street corner during a
year,
• the number of phone calls received by a switchboard each minute,
• the number of people who join a queue during a 5-minute time
interval.

Question 9

How is the Poisson distribution used to approximate binomial distribution?

Answer 9

• Conditions:
• The chance of success, p, is very small and may be unknown.
• The number of trials, n, is very large and again may be unknown.
• The average number of successes, μ = np, is known and constant and
is not large, say μ ≤ 7.
• Example: On average, a large airline loses half an aircraft a year
through accident. The number of planes lost each year follows a
Poisson distribution

Question 10

How can there be a Poisson distribution ?

Answer 10

• There are two conditions for the number of occurrences of an event
in a time interval to have a Poisson distribution:
• (i) The event occurs an average of μ times during the time interval of
interest. Events occur at the same average rate throughout the time
period.
• This means that the mean of a Poisson random variable is μ and so
on.
• (ii) Events happen independently and individually, so the number of
events that occur in any time interval is independent of the number
of events that occur in any other non-overlapping time interval.