Let X represent the number of minutes in a long-distance phone call. Suppose that X's density is given byf(x)=(110)e−x10;x&gt;0&nbsp;a) Verify that f is a density for a continuous random variable.&nbsp;b) Assuming that f adequately describes the behavior of the random variable X, find the probability that a randomly selected call will last at most 7 minutes; at least 7 minutes; exactly 7 minutes.&nbsp;c) Would it be ususual for a call to last between 1 and 2 minutes? Explain, based on the probability of this occuring.

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Let X represent the number of minutes in a long-distance phone call. Suppose that X&#039;s density is given byf(x)=(110)e−x10;x&amp;gt;0&amp;nbsp;a) Verify that f is a density for a continuous random variable.&amp;nbsp;b) Assuming that f adequately describes the behavior of the random variable X, find the probability that a randomly selected call will last at most 7 minutes; at least 7 minutes; exactly 7 minutes.&amp;nbsp;c) Would it be ususual for a call to last between 1 and 2 minutes? Explain, based on the probability of this occuring.

Sculd1987 · Accepted Answer

a. Let X be the length in minutes of a long-distance telephone conversation.Using the information provided, the density function for X isf(x)=110e−(110)x&amp;gt;0The following prerequisites need to be met in order to confirm that f is a density.f(x)&amp;gt;0Sum of probabilities should be equal to 1.All probabilities are greater than 0 according to the probability density function, which can be seen.∫0∞110e−(110)&amp;nbsp;dx&amp;nbsp;=110(e−(x10)(−110))0∞=−(e−(x10))0∞=−(e−∞−e0)=−(0−1)=1Thus, f is density function for a continuous random variable.Step 2b. The probability that a randomly selected call will last at most 7 minutes isP(X≤1)=P(0&amp;lt;X&amp;lt;1)=∫01110e−(x10)&amp;nbsp;dx&amp;nbsp;=110(e−(x10)(−110))01=−(e−(110)−e0)=−(0.9048−1)=0.0952The probability that a randomly selected call will last at least 7 minutes isP(X≥1)=1−P(X&amp;lt;1)=1−0.0952=0.9048Since the density function of X is continuous the probability that X equal to any particular value is 0. Thus, the probability that X equals to exactly 7 minutes is 0.Step 3c. The probability that a randomly selected call will last between 1 and 2 minutes isP(1&amp;lt;X&amp;lt;2)=∫12110e−(x10)&amp;nbsp;dx&amp;nbsp;=110(e−(x10)−110)12=−(e−(x10))12=−

Let X denote the length in minutes of a long-distance telephone conver

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