# Ch 5.1 Continuous Random Variable and Density Curve

## Ch 5.1 Continuous random variable

### A) Density Curve

Probability of a Continuous Random Variable X is  defined by its Probability Density Function(pdf) or density curve: $$f(x)$$ so that

- Area under the density curve corresponds to probability or relative frequency (percent).

- Total area under the density curve is equal to 1

- the graph is always above x-axis.

 Probability = Area = Percent

Two important continuous Probability Distributions

1) Uniform Distribution – The probability of X is equally likely to occur. Histogram of sample data usually bars of similar heights.  There is lowest and highest value of X.

Ex1.  X is modeled by Uniform Distribution for

lowest 2 and highest 8.8.

Probability that X is between 3 and 6 is the shaded area under the density curve.

### b) Normal Distribution

Probability that X is between a and b =

the area under the bell curve for x = a and x = b.

Shaded left area =

probability that x is less than a.

Shaded right area  =

probability that x is greater than a.

#### Notation and property of probability of Continuous random variable X.

Probability that X = a:  P(X = a) = 0

Probability that X is between a and b:

P( a < X < b) or  P(a ≤ X ≤ b)

Probability that X is less than a: P( X < a) = P( X ≤ a)

Probability that X is greater than a: P(X > a) = P(x ≥ a)