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:
- Compute the indefinite integral:
\[ \int \lambda e^{-\lambda x} , dx = -e^{-\lambda x} + C \]
- 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.
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