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STATLECT > Probability distributions > Exponential distribution

Exponential distribution - Exercise set 1

This exercise set contains some solved exercises on the computation of probabilities involving an exponential distribution. The theory needed to solve these exercises is introduced in the lecture entitled Exponential distribution.

Exercise 1.1

Let X be an exponential random variable with parameter [eq1]. Compute the following probability:[eq2]

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First of all we can write the probability as:[eq3]using the fact that the probability that an absolutely continuous random variable takes on any specific value is equal to zero (see Absolutely continuous random variables and zero-probability events). Now, the probability can be written in terms of the distribution function of X as:[eq4]

Exercise 1.2

Suppose the random variable X has an exponential distribution with parameter $\lambda =1$. Compute the following probability:[eq5]

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This probability can be easily computed using the distribution function of X:[eq6]

Exercise 1.3

What is the probability that a random variable X is less than its expected value, if X has an exponential distribution with parameter $lambda $?

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The expected value of an exponential random variable with parameter $lambda $ is:[eq7]The probability above can be computed using the distribution function of X:[eq8]

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