Chapter 5 - Continuous Distributions

Question 1

Probability density function

Answer 1

• measure of which values are more probable than others and to what degree
• analogous to the probability-mass function

Question 2

Cumulative distribution function

Answer 2

• the cumulative distribution function for the random variable X evaluated at the point a is defined as the probability that X will take on values less than or equal to a
• represented by the area under the pdf to the left of a

Question 3

Probability of individual values

Answer 3

• probability of individual values = 0

Question 4

Normal distribution

Answer 4

• most widely used continuous distribution
• also called the Gaussian distribution
• other distributions that are not normal can be made approximately normal by transforming data onto a different scale
• any random variable that can be expressed as a sum of many other random variables can be well approximated by a normal distribution

Question 5

The pdf of a normal distribution

Answer 5

• based on parameters of mean and standard deviation

- one standard deviation away from the mean represents the point of inflection

Question 6

Standard normal distribution

Answer 6

• a normal distribution with mean 0 and variance 1
• about 68% of the area under the standard normal density lies between +1 and -1
• about 95% of the area lies between +2 and -2
• about 99% of the area lies between +2.5 and 2.5

Question 7

Properties of the standard normal distribution

Answer 7

• a normal range for a biological quantity is often defined by a range within x standard deviations of the mean for some specified value of x

Question 8

Covariance

Answer 8

• if X and Y are independent, then the covariance between them is 0
• if large values of X and Y occur among the same subjects (or small values of X and Y), then the covariance is positive
• if large values of X and small values of Y(or conversely, small values of X and large values of Y) tend to occur among the same subjects, then the covariance is negative

Question 9

Correlation coefficient

Answer 9

• dimensionless quantity that ranges between -1 and 1
• if X and Y are approximately linearly related, a correlation coefficient of 0 implies independence
• correlation close to 1 implies nearly perfect positive dependence
• correlation close to -1 implies nearly perfect negative dependence

Question 10

Bernoulli trial

Answer 10

• random variable that takes on the value 1 with probability p and the value 0 with probability q=1-p
• special case of a binomial random variable with n = 1