Continuous Probability Distributions

• Continuous probability distributions are mathematical functions that describe the probabilities of possible results for a random variable that can take on any value in a continuous range, as opposed to a discrete distribution, which deals with discrete, or countable, outcomes.

• The graph of a continuous probability distribution is called the probability density function (PDF). The area under the curve of the PDF between two points corresponds to the probability that the random variable falls between those values. The total area under the PDF is always equal to 1.

• Important examples of continuous probability distributions used in Further Statistics 2 are the Normal distribution, the Exponential distribution, and the chi-square distribution.

• Normal distribution: This is commonly used due to its statistical properties. It is symmetrical, bell shaped and its mean, median and mode are all equal. The standard deviation measures the spread of the distribution.

• Normal distribution is characterised by two parameters: the mean (µ) which is the centre of distribution and the standard deviation (σ) which determines the spread of the distribution.

• Continuous uniform distribution: Is a type of continuous probability distribution in which all outcomes are equally likely. It has two parameters: the minimum value (a) and maximum value (b).

• Exponential distribution: This is the probability distribution of the time between events in an events that occur continuously and independently at a constant average rate. It has one parameter, the rate (λ).

• Each particular type of distribution has specific uses depending on the nature of the random variable and the data.

• Chi-Squared distribution: This distribution is applied, for example, in hypothesis testing or in construction of confidence intervals. It is based on the squared standard normal variable and the degrees of freedom.

• For each distribution, you should be able to calculate probabilities and percentiles, and conduct hypothesis tests. This often involves use of the cumulative distribution function (CDF).

• The cumulative distribution function (CDF) of a random variable is defined as the probability that the variable takes a value less than or equal to a certain value.

• Be aware of the Central Limit Theorem; it states that if you have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population with replacement, then the distribution of the sample mean will be approximately normally distributed.

• Learn how to use statistical tables (like the standard Normal table and the chi-square table) to find critical values and to calculate probabilities.

• Also, the use of technology (such as a graphical calculator with statistical functions or statistical software) is important as they offer an efficient way to calculate probabilities and critical values.