href="#ufb3522f6-20a5-586f-84c7-a5e056714a96">Section 7.3). It is always positive, but the exact shape depends on the degrees of freedom for the two chi‐squared distributions that determine it.
Uniform Distribution
The Uniform distribution (Figure 4.14d) has a rectangular shape so that each possible value occurs with equal probability within a given range. It can be useful in a Bayesian analysis as the prior distribution of an unknown parameter where all values with a given range are thought to be equally likely.
4.8 Points When Reading the Literature
1 What is the population from which the sample was taken? Are there any possible sources of bias that may affect the estimates of the population parameters?
2 Have reference ranges been calculated on a random sample of healthy volunteers? If not, how does this affect your interpretation? Is there any good reason why a random sample was not taken?
3 For any continuous variable, are the variables correctly assumed to have a Normal distribution? If not, how do the investigators take account of this?
4.9 Technical Section
Binomial Distribution
Data that can take only a 0 or 1 response, such as treatment failure or treatment success, follow the Binomial distribution provided the underlying population response rate π does not change. The Binomial probabilities are calculated from
for successive values of r from 0 through to n. In the above n! is read as n factorial and r! as r factorial. For r = 4, r! = 4 × 3 × 2 × 1 = 24. Both 0! and 1! are taken as equal to unity. It should be noted that the expected value for r, the number of successes yet to be observed if we treated n patients, is nπ. The potential variation about this expectation is expressed by the corresponding standard deviation
Poisson Distribution
Suppose events happen randomly and independently in time at a constant rate. If the events happen with a rate of λ events per unit time, the probability of r events happening in unit time is
where exp(−λ) is a convenient way of writing the exponential constant e raised to the power −λ. The constant e being the base of natural logarithms which is 2.718281…
The mean of the Poisson distribution for the number of events per unit time is simply the rate, λ. The variance of the Poisson distribution is also equal to λ, and so the SD = √λ.
Normal Distribution
The probability density, f(x), or the height of the curve above the x axis (see Figures 4.7 and 4.9) of a Normally distributed random variable x, is given by the expression,
where μ is the mean value of x and σ is the standard deviation of x. Note that for the Normal distribution π, is the mathematical constant 3.14159… and not the parameter of a Binomial distribution.
The probability density simplifies for the Standard Normal distribution, since μ = 0 and σ = 1, then the probability density, f(x), of a Normally distributed random variable x, is
4.10 Exercises
1 4.1 Which ONE of the following statements about probability is INCORRECT?If two binary outcomes (X and Y) are mutually exclusive the probability that either X or Y occurs is the sum of the probability that X occurs and the probability that Y occurs.If two binary outcomes (X and Y) are independent, then the probability that both outcomes X and Y occur is the probability that X occurs multiplied by the probability that Y occurs.When the outcome will definitely happen the probability of it happening is 1.When an outcome can never happen, the probability of it happening is 1.All probabilities range between 0 to 1.
2 4.2 Suppose we toss a single unbiased two‐sided coin three times in a row and record the number of heads. What is the probability of observing a head on three successive tosses?0.5000.2500.7500.1250.050
3 4.3 Suppose we roll a 10‐sided die (numbered 1 to 10) once.What is the probability of observing a score of 6 or more on this roll?0.40.50.60.70.08
4 4.4 Which ONE of the following statements about the Normal distribution is INCORRECT?The Normal distribution is symmetrical.The peak of the curve lies above the mean.The total area under the Normal distribution curve is 1.The mean and median will coincide.The bigger the variance the taller the peak.
5 4.5 Which (if any) of the following statements about the Normal distribution is CORRECT?Approximately 1% of the observations from a Normal distribution lie outside the mean ± 2SD. Approximately 2% of the observations from a Normal distribution lie outside the mean ± 2SD.Approximately 3% of the observations from a Normal distribution lie outside the mean ± 2SD.Approximately 5% of the observations from a Normal distribution lie outside the mean ± 2SD.Approximately 10% of the observations from a Normal distribution lie outside the mean ± 2SD
6 4.6 Which (if any) of the following statements about the Normal distribution is CORRECT?Approximately 1% of the observations from a Normal distribution lie outside the mean ± 1SD.Approximately 5% of the observations from a Normal distribution lie outside the mean ± 1SD.Approximately 10% of the observations from a Normal distribution lie outside the mean ± 1SD.Approximately 32% of the observations from a Normal distribution lie outside the mean ± 1SD.Approximately 95% of the observations from a Normal distribution lie outside the mean ± 1SD
7 4.7 The systolic blood pressure in the population of middle‐aged people is Normally distributed with a mean of 140 mmHg and a standard deviation of 10.What is the approximate probability of having a systolic blood pressure of 120 mmHg or less?0.0250.050.160.320.95What proportion of the population of middle‐aged people have a systolic blood pressure between 130 and 150 mmHg?0.950.680.320.160.90
8 4.8 The birthweight of babies in the population is Normally distributed
