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Chapter 3: Q5E (page 116)

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 is as follows:


Step by step solution

01

Given information

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


02

calculate the PDF

03

Sketch the PDF

In the graph, the x-axis denotes the random variable X and the y-axis denotes the corresponding PDF of X. For example, for X =1 corresponding PDF is 2/9S.

Plotting PDF of random variable X for every value of X, a PDF curve is obtained.

Then the PDF curve is given by

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Most popular questions from this chapter

Question:A certain drugstore has three public telephone booths. Fori=0, 1, 2, 3, let\({{\bf{p}}_{\bf{i}}}\)denote the probability that exactlyitelephone booths will be occupied on any Monday evening at 8:00 p.m.; and suppose that\({{\bf{p}}_{\bf{0}}}\)=0.1,\({{\bf{p}}_{\bf{1}}}\)=0.2,\({{\bf{p}}_{\bf{2}}}\)=0.4, and\({{\bf{p}}_{\bf{3}}}\)=0.3. LetXandYdenote the number of booths that will be occupied at 8:00 p.m. on two independent Monday evenings. Determine:

(a) the joint p.f. ofXandY;

(b) Pr(X=Y);

(c) Pr(X > Y ).

Suppose that the joint distribution of X and Y is uniform over a set A in the xy-plane. For which of the following sets A are X and Y independent?

a. A circle with a radius of 1 and with its center at the origin

b. A circle with a radius of 1 and with its center at the point (3,5)

c. A square with vertices at the four points (1,1), (1,鈭�1), (鈭�1,鈭�1), and (鈭�1,1)

d. A rectangle with vertices at the four points (0,0), (0,3), (1,3), and (1,0)

e. A square with vertices at the four points (0,0), (1,1),(0,2), and (鈭�1,1)

Show that there does not exist any numbercsuch that the following function would be a p.f.:

\(f\left( x \right) = \left\{ \begin{array}{l}\frac{c}{x}\;\;\;\;for\;x = 1,2,...\\0\;\;\;\;otherwise\end{array} \right.\)

Suppose that a fair coin is tossed 10 times independently.

Determine the p.f. of the number of heads that will be obtained.

Verify the rows of the transition matrix in Example 3.10.6 that correspond to current states\(\left\{ {AA,Aa} \right\}\)and\(\left\{ {Aa,aa} \right\}\)
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