Question

(please recall the last question from the quiz "Continuous and Discrete Random Variables and Probability Distribution" in which we discussed the relation between the probability and the area in case of a discrete random variable).

To find the probability that a random variable assumes a value between A and B, we will have to find the area bounded from the above by the density function, bounded from the below by the x-axis, and form the left and right by vertical lines through A and B.

Let's see how it works using the example above.

Find the probability that you call back during first 2 minutes. To that end you have to find the area shaded below. P(X<2) =

Answer 1

To find the probability that a random variable \( X \) assumes a value less than 2 (i.e., \( P(X < 2) \)), you typically need to know the probability density function (pdf) of the random variable. Given that we do not have a specific distribution defined in your prompt, I will provide a general approach using an example to illustrate how to compute this probability.

### Example: Exponential Distribution

Assume that the random variable \( X \) follows an exponential distribution with a rate parameter \( \lambda \). The probability density function (pdf) for an exponential distribution is given by:

\[
f(x) = \begin{cases}
\lambda e^{-\lambda x} & \text{for } x \geq 0 \\
0 & \text{for } x < 0
\end{cases}
\]

To find \( P(X < 2) \), you need to compute the area under the curve of the pdf from 0 to 2:

\[
P(X < 2) = \int_{0}^{2} f(x) \, dx
\]

Substituting in the pdf, we have:

\[
P(X < 2) = \int_{0}^{2} \lambda e^{-\lambda x} \, dx
\]

### Evaluating the Integral

To evaluate this integral:

1. Compute the indefinite integral:

\[
\int \lambda e^{-\lambda x} \, dx = -e^{-\lambda x} + C
\]

2. Evaluate it from 0 to 2:

\[
P(X < 2) = \left[-e^{-\lambda x}\right]_{0}^{2} = -e^{-2\lambda} - (-e^{0}) = 1 - e^{-2\lambda}
\]

### Final Probability Result

Thus, the probability that the random variable \( X \) is less than 2 is:

\[
P(X < 2) = 1 - e^{-2\lambda}
\]

### Conclusion

To conclude, to find \( P(X < 2) \), we find the area under the density function curve \( f(x) \) between \( x = 0 \) and \( x = 2 \). If you have a specific probability density function you want to use, simply apply the same concept by performing the definite integral of the pdf from the lower bound to the upper bound.