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Statistical analysis Using R and Excel (SURE)

Chapter 05 Continuous Probability Distributions

( For a discrete random variable , the probability function f(x) provides the probability that the random variable assumes a particular value . With a continuous random variable , the probability function or probability density function doesn ’ t directly provide the probability and the area under the graphs of f(x) corresponding to a given interval does provide the probability that the continuous random variable x assumes a value in that interval. probability mass function ( p.m.f .) ↔ probability density function ( p.d.f .) )

1. Uniform Probability Distribution

1) Definition

- A Continuous probability distribution for which the probability that the random variable will assume a value in any interval is the same for each interval of equal length.

2) Uniform Probability Density Function

①

② E(x) = (a + b) / 2, Var(x) = (b – a)2 / 12

2. Normal Probability Distribution

1) Definition

- A Continuous probability distribution whose probability density function is bell shaped and determined by its mean μ and standard deviation σ; X ~ N(μ, σ2)

2) Probability Density Function

①

② f(x) ≥ 0 and

③ E(x) = μ and Var(x) = σ2

3) Properties of Normal Distribution

① The entire family of normal distribution is differentiated by two parameters, μ and σ.

② The highest point on the normal curve is at the mean. (mean = median = mode)

③ The normal distribution is symmetric about the mean.

④ The standard deviation determines how flat and wide the normal curve is.

⑤ The percentages of values in some commonly used intervals are as follows.

4) Standard Normal Distribution

When X ~ N(μ, σ2), is normally distributed with a mean of 0 and a standard deviation of 1, that is Z ~ N(0, 12).

. E(x) = 0 and Var(x) = 1.

For the standard normal distribution, areas under the normal curve can be available in tables that can be used to compute probabilities.

3. Exponential Probability Distribution

1) Definition

- Used for random variables such as the time between arrivals at a car wash, the time required to load a truck, the time until the failure of a machine, the waiting time in service line, distance between major defects in a highway, and so on.

2) Exponential Probability Density Function

①

② E(x) = μ and Var(x) = μ2

③ F(x0) = P(x ≤ x0) =

3) Relationship with Poisson Distribution

If the Poisson distribution provides an appropriate description of the number of occurrences per interval, the exponential distribution provides a description of the length of the interval between occurrences.