Probability Distributions and Random Variables

Question 1

Binomial Distribution

How many parameters does it have and what are they?

What three things can be said about the Binomial distribution?

Answer 1

The Binomial Distribution has two parameters:

1) Number of trials, n.
2) Probability of outcome occurring, P.

1) The underlying probability experiment has two possible outcomes.
2) Probability P does not change between trials.
3) Distribution appropriate for any value of P between 0 and 1.

Question 2

What is a random variable?

Answer 2

A random variable is a variable whose value or outcome is the result of chance and is therefore unpredictable, although the range of possible outcomes and the probability of each outcome may be known.

Question 3

The Normal Distribution

When does it arise?

What five things can be said about the Normal Distribution?

Answer 3

The normal distribution tends to arise when a random variable is the result of many independent random influences added together, none of which dominates the others.

1) It applies to continuous random variables such as height - Can be evaluated for any values of x.
3) Unimodal - single central peak.
4) Symmetric
5) Bell-shaped
5) Y-axis labelled f(x) - area under curve represents probability.

Question 4

How many parameters does the normal distribution have and what are they?

Answer 4

The normal distribution has two paramaters:

1) Mean, μ
2) Standard Deviation, σ

Question 5

What is the mean and variance of the standard normal distribution?

Answer 5

Mean = 0
Variance = 1

Question 6

What is a z-score?

Answer 6

The z-score is used to transform data so that they accord with the standard normal distribution by shifting the original distribution μ units to the left and adjusting dispersion by dividing through by σ, resulting in a mean of 0 and variance of 1.

Question 7

The sample mean is normally distributed under what condition?

Answer 7

As long as observations are independently drawn, the sample mean is normally distributed.

Question 8

Sampling from a non-normal distribution

What is the Central Limit Theorem?

At what point is the approximation said to appropriate?

Answer 8

The mean of a random sample drawn from a population with mean, μ and variance, σ (squared), has a sampling distribution which approaches a normal distribution as the sample size approaches infinity.

Approximation is appropriate if n > 25.

Question 9

The Poisson Distribution

When is it used?

Answer 9

1) Used when P is very small and nP is less than 5.
2) Used to approximate binomial.
3) Used in problems where events occur over time.
4) Used when n and P cannot be identified separately.