Week 07 - Distribution for a continuous random variable

probability density function

A function used to compute probabilities for a continuous random variable. The area under the graph of a probability density function over an interval represents probability.

• the total area should always be equal to 1

• should not be negative

• not a probability

density curve

A mathematical model used to describe the overall pattern of the distribution of a random variable.

cumulative distribution function

A function giving the probability that a random variable is less than or equal to a specified value.
- take the integral from -inf to x of P(X

f(x) vs F(x)

the lower case is a pmf/pdf while the upper is a cdf

uniform distribution

Distribution where populations are spaced evenly, like a line.

height of uniform distribution

1/(b-a)

E(X) of a pdf

x (or function) times the integral of the pdf from -inf to inf

V(X) of a pdf

is the same as E(X^2) - E(X)^2 but with integrals instead of summation as it is not discrete.

normal distribution

A function that represents the distribution of variables as a symmetrical bell-shaped graph.
mean \= median \= mode and always approaches 0 in tails

p-th percentile

the value (Xp), such that p% of the measurements will fall below that value and (100-p)% of the measurements will fall above that value

standardized value

value found by subtracting the mean and dividing by the standard deviation

68-95-99.7 rule

in a normal model, about 68% of values fall within 1 standard deviation of the mean, about 95% fall within 2 standard deviations of the mean, and about 99.7% fall within 3 standard deviations of the mean

E(X) for a normal distribution

is the E(X) of pdf

V(X) for a normal distribution

is the same of a variance of a pdf

Normal Approximation

Suppose that a count x of successes has the binomial distribution with n trials and success probability p. When n is large, the distribution of X has a mean np and standard deviation √(np(1-p)).

Continuity correction

Adjustment made when a discrete random variable is being approximated by a continuous random variable