Download Lecture Note, July 14, 2014 Chih-Hsin Hsueh I. Binomial Distribution

Lecture Note, July 14, 2014
Chih-Hsin Hsueh
I. Binomial Distribution – Example It is known that screws produced by a certain
company will be defective with probability .01 independently of each other. The company
sells the screws in packages of 10 and offers a money-back guarantee that at most 1 of the 10
screws is defective. What proportion of packages sold must the company replace?
II. Continuous Random Variable
• Probability density function
– The probability distribution of a continuous random variable is known as a continuous probability distribution. It can be described by a curve, called density
curve.
– Properties of density curve
1. The probability under density curve y = f (x) is equal to 1.
2. P (a ≤ X ≤ b) = area under curve between a and b.
3. F (a) = P (X ≤ a) is called the cumulative distribution function (cdf) of the
random variable X, which is the area under the curve to the left of a.
4. 0 ≤ F (X) ≤ a and F (X) increases with a.
• Mean and Variance for continuous random variable: Let X be a continuous random variable with probability distribution f (x). Then the mean (expectation/expected value),
variance and standard deviation of X are given by the following :
Z
µ = E(X) = xf (x)dx
Z
2
σ = V ar(X) = (x − µ)2 f (x)dx
√
σ = Sd(X) = σ 2
1
• Some facts:
III. Normal Distribution
• It is a special type of density curve which describes a particular type of continuous probability distribution, called a normal distribution. These density curves are symmetric,
unimodal(one peak) and bell-shaped.
• It is one type of continuous probability distribution, which is described by a normal
curve. It is a two- parameter distribution (the mean, µ, and the standard deviation, σ).
This distribution is very much useful and has a wide application in the field of statistical
inference.
• Density function, mean, variance for Normal distribution:
• Standard Normal distribution: A normal distribution with mean 0 and standard deviation 1 is called the standard normal distribution, denoted by N (0, 1)
• Any normal random variable X with mean and standard deviation can be transformed
into a random variable Z having a standard normal distribution via the standardization:
z=
2
X −µ
σ
3