Continuous probability distributions

 

A continuous random variable X has a density function given by: \[ \begin{align} f(x) &= 2e^{-2x} \qquad x \ge 0 \\ &= 0 \qquad \text{elsewhere} \end{align}\] Find the probability of X taking a value between 0 and 3, given it has taken a value less than 4.

Calculate the probability of x taking a value in the larger interval.

\[ \begin{align} \text{Pr}(X \le 4) &= \int_0^4 2e^{-2x} dx \\ &= [ -e^{-2x}]_0^4 \\ &= -e^{-8} - (-e^0) \\ &= 1 - e^{-8} \end{align} \ \]

Calculate the probability of x taking a value in the smaller interval

\[ \begin{align} \text{Pr}(X \le 3) &= \int_0^3 2e^{-2x} dx \\ &= [ -e^{-2x}]_0^3 \\ &= -e^{-6} - (-e^0) \\ &= 1 - e^{-6} \end{align} \ \]

Use the expression for conditional probabilty

\[ \begin{align} \text{Pr}(0 \le X \le 3 | X \le 4) &= \frac{\text{Pr}(X \le 3)}{\text{Pr}(X \le 4)} \\ &= \frac{ 1 - e^{-6}}{1 - e^{-8}} \end{align} \ \]

You will gain most benefit if you work through each problem first and then come back and check your work against the solution.

 

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Now You Try:

A continuous random variable X has a density function given by:
\[ \begin{align} f(x) &= \frac{1}{3}e^{-\frac{x}{3}} \qquad x \ge 0 \\ &= 0 \qquad \text{ elsewhere} \end{align}\] Find the probability of X taking a value between 0 and 2, given it has taken a value less than 3.