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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鈥�(x)

f (x) = ex-3 where x鈮� 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

Question:Suppose thatXandYare random variables such that(X, Y)must belong to the rectangle in thexy-plane containing all points(x, y)for which 0鈮�x鈮�3 and 0鈮�y鈮�4. Suppose also that the joint c.d.f. ofXandYat every point

(x,y) in this rectangle is specified as follows:

\({\bf{F}}\left( {{\bf{x,y}}} \right){\bf{ = }}\frac{{\bf{1}}}{{{\bf{156}}}}{\bf{xy}}\left( {{{\bf{x}}^{\bf{2}}}{\bf{ + y}}} \right)\)

Determine

(a) Pr(1鈮�X鈮�2 and 1鈮�Y鈮�2);

(b) Pr(2鈮�X鈮�4 and 2鈮�Y鈮�4);

(c) the c.d.f. ofY;

(d) the joint p.d.f. ofXandY;

(e) Pr(Y鈮�X).

Suppose that a random variableXhas a discrete distribution

with the following p.f.:

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

Find the value of the constantc.

Suppose that a coin is tossed repeatedly until a head is obtained for the first time, and let X denote the number of tosses that are required. Sketch the c.d.f of X.

Suppose that the c.d.f.Fof a random variableXis as sketched in Fig. 3.9. Find each of the following probabilities:

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

f(x)={ \begin{aligned}c{e^{-2x}}\\0\end{aligned}

for x > 0, otherwise.

a. Find the value of the constant c and sketch the p.d.f.

b. Find the value of Pr (1 <X< 2)
See all solutions

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