For continuous probability distributions, PROBABILITY = AREA.

μ=μX=E[X]=∞∫−∞x⋅f(x)dx. The formula for the expected value of a continuous random variable is the continuous analog of the expected value of a discrete random variable, where instead of summing over all possible values we integrate (recall Sections 3.6 & 3.7).

Let X be a positive continuous random variable. Prove that EX=∫∞0P(X≥x)dx….Solution

1. To find c, we can use ∫∞−∞fX(u)du=1: =∫∞−∞fX(u)du. =∫1−1cu2du.
2. To find EX, we can write. EX. =∫1−1ufX(u)du.
3. To find P(X≥12), we can write P(X≥12)=32∫112x2dx=716.

Continuous variables are variables whose value between two values is infinite. They can be numerical, time or date. Some examples are temperature and length.

Continuous probability distribution: A probability distribution in which the random variable X can take on any value (is continuous). Because there are infinite values that X could assume, the probability of X taking on any one specific value is zero. The normal distribution is one example of a continuous distribution.

1. Consider the function f(x) = for 0 ≤ x ≤ 20.
2. f(x) =
3. The graph of f(x) =
4. The area between f(x) = where 0 ≤ x ≤ 20 and the x-axis is the area of a rectangle with base = 20 and height = .
5. Suppose we want to find P(x = 15).
6. Label the graph with f(x) and x.

Continuous data is data that can take any value. Height, weight, temperature and length are all examples of continuous data. Some continuous data will change over time; the weight of a baby in its first year or the temperature in a room throughout the day.

This is a useful fact. X is a continuous random variable with probability density function given by f (x) = cx for 0 ≤ x ≤ 1, where c is a constant. Find c. If we integrate f (x) between 0 and 1 we get c/2. Hence c/2 = 1 (from the useful fact above!), giving c = 2.

Let X be a continuous random variable with PDF fX(x) = {x2(2x + 3 2) 0 < x ≤ 1 0 otherwise If Y = 2 X + 3, find Var (Y). First, note that Var ( Y) = Var ( 2 X + 3) = 4 Var ( 1 X), using Equation 4.4 Thus, it suffices to find Var ( 1 X) = E [ 1 X 2] − ( E [ 1 X]) 2.

For any continuous random variable with probability density function f (x), we have that: This is a useful fact. X is a continuous random variable with probability density function given by f (x) = cx for 0 ≤ x ≤ 1, where c is a constant. Find c. If we integrate f (x) between 0 and 1 we get c/2.

Discrete and Continuous Random Variables: A variable is a quantity whose value changes. A discrete variable is a variable whose value is obtained by counting. Examples: number of students present