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Chapter 3: Q6E (page 117)

Suppose that the c.d.f. of a random variable X is as follows:

Find and sketch the p.d.f. of X

Short Answer

Expert verified

The c.d.f. of a random variable X is given as follows:

Step by step solution

01

Given the information

The c.d.f. of a random variable X is given as follows:

02

Calculating the PDF

f(x) = F’(x)

f (x) = ex-3 where x≤ 3

03

Sketch the PDF

In the graph, x-axis denotes the values of random variable x and the y-axis denotes its corresponding pdf values.

For example, for X= 2.5, PDF value = 0.60653. Plotting all the values of PDF corresponding to every values of X , the PDF curve is drawn.

Thus, the PDF curve is given by

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Most popular questions from this chapter

In a large collection of coins, the probability X that a head will be obtained when a coin is tossed varies from one coin to another, and the distribution of X in the collection is specified by the following p.d.f.:

\({{\bf{f}}_{\bf{1}}}\left( {\bf{x}} \right){\bf{ = }}\left\{ {\begin{align}{}{{\bf{6x}}\left( {{\bf{1 - x}}} \right)}&{{\bf{for}}\,{\bf{0 < x < 1}}}\\{\bf{0}}&{{\bf{otherwise}}}\end{align}} \right.\)

Suppose that a coin is selected at random from the collection and tossed once, and that a head is obtained. Determine the conditional p.d.f. of X for this coin.

Question:In Example 3.4.5, compute the probability that water demandXis greater than electric demandY.

Suppose that the n variables\({{\bf{X}}_{\bf{1}}}{\bf{ \ldots }}{{\bf{X}}_{\bf{n}}}\) form a random sample from the uniform distribution on the interval [0, 1]and that the random variables \({{\bf{Y}}_{\bf{1}}}\;{\bf{and}}\;{{\bf{Y}}_{\bf{n}}}\) are defined as in Eq. (3.9.8). Determine the value of \({\bf{Pr}}\left( {{{\bf{Y}}_{\bf{1}}} \le {\bf{0}}{\bf{.1}}\;{\bf{and}}\;{\bf{Y}}_{\bf{n}}^{} \le {\bf{0}}{\bf{.8}}} \right)\)

Suppose that a coin is tossed repeatedly in such a way that heads and tails are equally likely to appear on any given toss and that all tosses are independent, with the following exception: Whenever either three heads or three tails have been obtained on three successive tosses, then the outcome of the next toss is always of the opposite type. At time\(n\left( {n \ge 3} \right)\)let the state of this process be specified by the outcomes on tosses\(n - 2\),\(n - 1\)and n. Show that this process is a Markov chain with stationary transition probabilities and construct the transition matrix.

Let W denote the range of a random sample of nobservations from the uniform distribution on the interval[0, 1]. Determine the value of

\({\bf{Pr}}\left( {{\bf{W > 0}}{\bf{.9}}} \right)\).
See all solutions

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