C2: Discrete Probability Distributions

Question 1

Variable

Answer 1

= that characteristic in a popn of interest that changes from subject to subject.
eg. Age ; Height.

Question 2

Random variable

Answer 2

= a characteristic whose value is unknown until that value is observed (due tonthe result of an experiment).
• value varies from trial to trial (eg. Flipping a coin).
• has a probability distr. in a popn context.

Question 3

Types Of Random Variables(2)

Answer 3

[A] Discrete random variable
= numerical variable that can take on discrete values.
• found by counting.
• has characteristics of whole no.
eg. 1, 2, 3.

[B] Continuous random variable
= numerical variable that can take on any numerical value.
• found by measuring.
• has characteristics of decimal no.
eg. 1.5, 2.7

Question 4

Binomial Theorem

Answer 4

(p + q)^n = SUM OF nCr p^x q^(n-x)

Where:
p -- possible terms -- P(success)
q -- possible terms -- P(failure)
n -- no. of trial.
nCr -- binomial coefficient.
x -- no. of times one has P(success) and P(failure).

Question 5

Binomial Distribution

Answer 5

• fixed no. of trials = n.
• independent trials with identical conditions.
• 2 outcomes = success & failure.
• fixed probability of success = p & q.
• Mean = np
• Variance = npq

● P(X) = nCr • p^r • q^(n-r)

☆ nCr = n! / r!(n-r)!

Question 6

Poison Distribution

Answer 6

= events occur at random points in time, space or volume.
• most widely used.

• Mean = Variance = lambda

● P(X=x) = e^(-lambda) / x!

Where:

• lambda = average no. of items per unit space/time.
• e = Euler’s no. = 2.71828
• x = no. of occurrences.

Question 7

Poisson Approximation To The Binomial Distribution

Answer 7

• Conditions:

• n –> infinity (becomes larger).
• p –> 0 (becomes smaller).

● Binomial distribution formula ~ Poisson distribution formula