Statistics -Continuous probability distribution

1. Statistics-Continuous probability distribution 2013/11/18

2. Probability density function • With continuous ransom variables, the counterpart of the probability function is the probability density function, denoted by f(x) <Note> How to compute Pr(a≤x≤b)? <Note>The probability of any particular value of the continuous random variable is zero.

3. Continuous probability distribution • For a continuous random variable x: • The probability distribution is defined by a probability density function, denoted by f(x) • The expected value of a continuous random variable is a measure of the central location for the random variable. • The variance is used to summarize the variability in the values of a random variable.

4. Uniform probability distribution • Uniform probability density function: • Expected value for uniform probability distribution: • Variance for uniform probability distribution: f (x) = 1/(b – a) for a<x<b = 0 elsewhere E(x) = (a + b)/2 Var(x) = (b - a)2/12

5. Normal probability distribution • Normal probability density function: • Expected value for normal probability distribution: • Variance for normal probability distribution:

6. Standard normal probability distribution • Standard normal probability density function: • Expected value for standard normal probability distribution: 0 • Variance for standard normal probability distribution: 1

7. Exponential probability distribution • Exponential probability density function: • Expected value for exponential probability distribution: • Variance for exponential probability distribution:

8. Other distributions • Chi-square distribution • t distribution • F distribution • others

9. Relationships between distributions • Normal distribution vs. Standard normal distribution • Normal distribution vs. Binomial distribution • Poisson distribution vs. Exponential distribution