ECON1203 Lecture Notes - Lecture 6: Probability Density Function, Probability Distribution, Random Variable
5 – Continuous Probability
Continuous random variables
• Probability density function (pdf)
Continuous version of probability histogram used for discrete random variables
Pdf for otiuous rado ariale X, ith rage a ≤ ≤ , must satisfy:
(a) Fx ≥ 0 for all x between a and b
(b) Total area under the curve between a and b is unity (i.e. equal to 1)
Probabilities are now represented by areas under the pdf
Uniform random variable
• Uniform pdf:
• Cumulative density function (cdf):
Cdf for a random variable is defined as:
(a) F = P X ≤
Often called the distribution function of X
The normal distribution
• For a normally distributed random variable:
P (X = x) = 0 **
P (a < X < b) = area under pdf curve, between possible values a and b **
(a) ** = true for continuous random variables
A normal distribution is completely characterised by its mean, µ, and variance, 2
• Graphically, normal probability density function (pdf) is symmetric, unimodal and
bell-shaped
Mean = median = mode
• Basic features:
find more resources at oneclass.com
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Document Summary
Continuous random variables: probability density function (pdf) Continuous version of probability histogram used for discrete random variables. Probabilities are now represented by areas under the pdf. Uniform random variable: uniform pdf, cumulative density function (cdf): Cdf for a random variable is defined as: (a) f(cid:894)(cid:454)(cid:895) = p (cid:894)x (cid:454)(cid:895) Often called the distribution function of x. The normal distribution: for a normally distributed random variable: P (x = x) = 0 ** P (a < x < b) = area under pdf curve, between possible values a and b ** (a) ** = true for continuous random variables. A normal distribution is completely characterised by its mean, , and variance, 2: graphically, normal probability density function (pdf) is symmetric, unimodal and bell-shaped. Mean = median = mode: basic features: Range of support is unlimited: (cid:454) . Despite u(cid:374)li(cid:373)ited ra(cid:374)ge, there is little pro(cid:271)a(cid:271)ilit(cid:455) area i(cid:374) the (cid:858)tails(cid:859) (a) 4. 6% outside 2 (b) 0. 3% outside 3 .
