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Chapter 4: Problem 5

Let \(Y_{1}

Short Answer

Expert verified
The probability that the maximum of four values, each selected randomly and independently from an exponential distribution with parameter 1, is at least 3 is \( (1 - e^{-3})^4 \).

Step by step solution

01

Identify and understand the given function

The given function is \(f(x) = e^{-x}\), which is a pdf for the continuous exponential distribution with parameter \(1\). We verify that the function is indeed a pdf by confirming that it is non-negative and that its integral over all possible values of \(x\) equals \(1\).
02

Find the cumulative distribution function (CDF)

The CDF of a random variable is the probability that the variable takes a value less than or equal to a specific value. It is found by integrating the pdf from \(-\infty\) to a specific value \(x\). In this case, we integrate the given function from \(0\) to \(x\), resulting in \(F(x) = 1 - e^{-x}\).
03

Use the CDF to find the probability

To find \(P(Y_4 \geq 3)\), we need to find the probability that all values \(Y_1, Y_2, Y_3, Y_4\) are less than \(3\). Since the samples were selected independently, the overall probability is the product of the individual probabilities, which are all the same due to the same distribution and hence is \(F(3)^4 = (1 - e^{-3})^4\).

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