Question

Consider a random sample of size 4 from an exponential distribution with parameter θ = 1, so fX(x) = e^(-x) for 0 < x. (a) Find P(Y1 ≤ 1). (b) Find P(Y4 ≥ 3)."

Answer 1

A random sample of size 4 from an exponential distribution with parameter θ = 1, so fX(x) = e^(-x) for 0 < x is:

(a) P(Y1 ≤ 1) = 1 - e^(-1)

(b) P(Y4 ≥ 3) = 1 - e^(-3)

Step-by-step explanation:

In part (a), to find P(Y1 ≤ 1), we use the cumulative distribution function (CDF) of the exponential distribution. The CDF for an exponential distribution with parameter θ is given by F(x) = 1 - e^(-x/θ). In this case, θ = 1, so the CDF becomes F(x) = 1 - e^(-x). Therefore, P(Y1 ≤ 1) is calculated by substituting x = 1 into the CDF, resulting in the final answer 1 - e^(-1).

Moving on to part (b), to find P(Y4 ≥ 3), we use the complementary probability. The event Y4 ≥ 3 is the same as 1 minus the event Y4 < 3. Using the CDF, P(Y4 < 3) can be calculated by substituting x = 3 into 1 - e^(-x), yielding e^(-3). Therefore, P(Y4 ≥ 3) is 1 - e^(-3).

In summary, part (a) involves finding the probability of the random variable being less than or equal to a specific value, while part (b) involves finding the probability of the random variable being greater than or equal to another specific value. Both calculations utilize the properties of the exponential distribution and its cumulative distribution function.