Continuous Probability Distributions

• A continuous random variable has a probability of zero of assuming exactly any of its values.
  $$P(a<X \leq b) = P(a<X<b) = P(a \leq X <b) = P(a \leq X \leq b)$$

• Example: Height of a random person. $P(X=178 cm)=0$. No assuming exactly.
• With continuous random variables we talk about the probability of $x$ being in some interval, like $P(a< X < b)$, rather than $x$ assuming a precise value like $P(X=a)$.
• Its probability distribution cannot be given in tabular form, but can be stated as a formula, a function of the numerical values of the continuous random variables.
• Some of these functions are shown below:

Figure 3.3: Typical density functions.

• Definition 3.6:
  

A probability density function is constructed so that the area under its curve bounded by the $x$ axis is equal to 1.

Figure 3.4: $P(a< X < b)$

• Example 3.11: Suppose that the error in reaction temperature in $^\circ$C is a continuous random variable $X$ having the probability density function
  $$f(x)=\left\lbrace \begin{array}{l} \frac{x^2}{3}~ for~ -1 < x < 2 \\ 0, ~elsewhere \\ \end{array}\right\rbrace$$

• Verify $\int_{-\infty}^\infty f(x)dx=1$
  • $\int_{-\infty}^\infty f(x)dx=\int_{-1}^2 \frac{x^2}{3}dx=\frac{x^3}{9}\textbar_{-1}^2=\frac{8}{9}+ \frac{1}{9}=1$
• Find $P(0< X < 1)$
  • $P(0< X < 1)=\int_0^1 \frac{x^2}{3}dx=\frac{x^3}{9}\textbar_0^1=\frac{1}{9}$

• Definition 3.7:
  
• An immediate consequence:
  • $P(a < X < b) = F(b) -F(a)$
  • $f(x) = \frac{dF(x}{dx})$, if the derivative exists

Example 3.12: For the density function of Example 3.6 find $F(x)$, and use it to evaluate $P(0 < X \leq 1)$.

For $-1 < x < 2$

$$F(x)=\int_{-\infty}^x f(t)dt=\int_{-\infty}^x \frac{t^2}{3}dt$$

$$=\frac{t^3}{9}\textbar_{-1}^x=\frac{x^3+1}{9}$$

$$F(x)=\left\lbrace \begin{array}{l} 0,~x \leq -1 \\ \\ \f... ...\leq x <2 \\ \\ 1, x \geq 2 \\ \\ \end{array}\right\rbrace$$

$$P(0<X\leq 1)=F(1)-F(0)$$

$$=\frac{2}{9}-\frac{1}{9}=\frac{1}{9}$$

Figure 3.5: Continuous cumulative distribution function.