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

2. Probability density function f ( x ) With continuous ransom variables, the counterpart of the probability function is the probability density function, denoted by f ( x ) How to compute Pr(a≤x≤b)? How to compute Pr(a≤x≤b)? The probability of any particular value of the continuous random variable is zero. 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 f ( x ) = 1/( b – a ) for a < x < b = 0 elsewhere = 0 elsewhere f ( x ) = 1/( b – a ) for a < x < b f ( x ) = 1/( b – a ) for a < x < b = 0 elsewhere = 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