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Chapter 3: Q3E (page 129)

Suppose thatXandYhave a discrete joint distribution for which the joint p.f. is defined as follows:


Short Answer

Expert verified

a. The value of the constant is 0.025.

b. The probability is 0.05.

c. The probability is 0.175.

d. The probability is 0.70.

Step by step solution

01

Given the information

02

Finding the value of constant

The value of the constant is 0.025.

The joint pmf of X and Y is given by,


03

Calculating the probability for part (b)

b.

Pr (X=0 and Y = -2) = 0.025|0-2|

= 0.05

The probability is 0.05

04

Calculating the probability for part (c)

05

Calculating the probability for part (d)

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Most popular questions from this chapter

Start with the joint distribution of treatment group and response in Table 3.6 on page 138. For each treatment group, compute the conditional distribution of response given the treatment group. Do they appear to be very similar or quite different?

Suppose that the joint p.d.f. of two random variables X and Y is as follows:

\(f\left( {x,y} \right) = \left\{ \begin{aligned}{l}c\left( {x + {y^2}} \right)\,\,\,\,\,\,for\,0 \le x \le 1\,and\,0 \le y \le 1\\0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,otherwise\end{aligned} \right.\)

Determine

(a) the conditional p.d.f. of X for every given value of Y, and

(b) \({\rm P}\left( {X > \frac{1}{2}|Y = \frac{3}{2}} \right)\).

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 ).

Consider the situation described in Example 3.7.14. Suppose that \(\) \({X_1} = 5\) and\({X_2} = 7\)are observed.

a. Compute the conditional p.d.f. of \({X_3}\) given \(\left( {{X_1},{X_2}} \right) = \left( {5,7} \right)\).

b. Find the conditional probability that \({X_3} > 3\)given \(\left( {{X_1},{X_2}} \right) = \left( {5,7} \right)\)and compare it to the value of \(P\left( {{X_3} > 3} \right)\)found in Example 3.7.9. Can you suggest a reason why the conditional probability should be higher than the marginal probability?

Let X have the uniform distribution on the interval, and let prove that \({\bf{cX + d}}\) it has a uniform distribution on the interval \(\left[ {{\bf{ca + d,cb + d}}} \right]\)
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