2. In conducting an experiment in the laboratory, temperature gauges are to be used at four junction points in the equipment setup. These four gauges are ran- domly selected from a bin containing seven such gauges. Unknown to the sci- entist, three of the seven gauges give improper temperature readings. Let X denote the number of defective gauges selected and Y.the number of nondefec- tive gauges selected. The joint density for (X, Y) is given in Table 5.5. (a) The values given in Table 5.5 can be derived by realizing that the random variable X is hypergeometric. Use the results of Sec. 3.7 to verify the val- ues given in Table 5.5. (b) Find the marginal densities for both X and Y. What type of random variable is Y? (c) Intuitively speaking, are X and Y independent? Justify your answer mathe- matically. 3. The joint density for (X, Y) is given by fxy (x, y) = 1/n² x = 1, 2, 3, . ..,n y = 1, 2, 3, . .. , n (a) Verify that fxy (x, y) satisfies the conditions necessary to be a density. (b) Find the marginal densities for X and Y. (c) Are X and Y independent? 4. The joint density for (X, Y) is given by fxy (x, y) = 2/n(n + 1) 1
2. In conducting an experiment in the laboratory, temperature gauges are to be used at four junction points in the equipment setup. These four gauges are ran- domly selected from a bin containing seven such gauges. Unknown to the sci- entist, three of the seven gauges give improper temperature readings. Let X denote the number of defective gauges selected and Y.the number of nondefec- tive gauges selected. The joint density for (X, Y) is given in Table 5.5. (a) The values given in Table 5.5 can be derived by realizing that the random variable X is hypergeometric. Use the results of Sec. 3.7 to verify the val- ues given in Table 5.5. (b) Find the marginal densities for both X and Y. What type of random variable is Y? (c) Intuitively speaking, are X and Y independent? Justify your answer mathe- matically. 3. The joint density for (X, Y) is given by fxy (x, y) = 1/n² x = 1, 2, 3, . ..,n y = 1, 2, 3, . .. , n (a) Verify that fxy (x, y) satisfies the conditions necessary to be a density. (b) Find the marginal densities for X and Y. (c) Are X and Y independent? 4. The joint density for (X, Y) is given by fxy (x, y) = 2/n(n + 1) 1
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Binomial Distribution
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![2. In conducting an experiment in the laboratory, temperature gauges are to be
used at four junction points in the equipment setup. These four gauges are ran-
domly selected from a bin containing seven such gauges. Unknown to the sci-
entist, three of the seven gauges give improper temperature readings. Let X
denote the number of defective gauges selected and Y.the number of nondefec-
tive gauges selected. The joint density for (X, Y) is given in Table 5.5.
(a) The values given in Table 5.5 can be derived by realizing that the random
variable X is hypergeometric. Use the results of Sec. 3.7 to verify the val-
ues given in Table 5.5.
(b) Find the marginal densities for both X and Y. What type of random variable
is Y?
(c) Intuitively speaking, are X and Y independent? Justify your answer mathe-
matically.
3. The joint density for (X, Y) is given by
fxy (x, y) = 1/n²
x = 1, 2, 3, . ..,n
y = 1, 2, 3, . .. , n
(a) Verify that fxy (x, y) satisfies the conditions necessary to be a density.
(b) Find the marginal densities for X and Y.
(c) Are X and Y independent?
4. The joint density for (X, Y) is given by
fxy (x, y) = 2/n(n + 1)
1<y<x<n
n a positive integer
(a) Verify that fyy (x, y) satisfies the conditions necessary to be a density. Hir
The sum of the first n integers is given by n(n + 1) /2.d
(b) Find the marginal densities for X and Y. Hint: Draw a picture of the regi
over which (X, Y ) is defined.
(c) Are X and Y independent?
(d) Assume that n = 5. Use the joint density to find P[X< 3 and Y< 2]. Fï
P[X < 3] and P[Y < 2]. Hint: Draw a picture of the region over wh
(X, Y) is defined.
S. The two most common types of errors made by programmers are syntax err
and errors in logic. For a simple language such as BASIC the number of s
errors is usually small. Let X denote the number of syntax errors and Y
number of errors in logic made on the first run of a BASIC program. Assu
that the joint density for (X, Y) is as shown in Table 5.6.
3.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc074fc7e-4e48-46f1-bfc4-aee6ba4e6262%2Ff5dc772d-caaf-42a8-a48d-7fba3c11a494%2Fqib6qkf.jpeg&w=3840&q=75)
Transcribed Image Text:2. In conducting an experiment in the laboratory, temperature gauges are to be
used at four junction points in the equipment setup. These four gauges are ran-
domly selected from a bin containing seven such gauges. Unknown to the sci-
entist, three of the seven gauges give improper temperature readings. Let X
denote the number of defective gauges selected and Y.the number of nondefec-
tive gauges selected. The joint density for (X, Y) is given in Table 5.5.
(a) The values given in Table 5.5 can be derived by realizing that the random
variable X is hypergeometric. Use the results of Sec. 3.7 to verify the val-
ues given in Table 5.5.
(b) Find the marginal densities for both X and Y. What type of random variable
is Y?
(c) Intuitively speaking, are X and Y independent? Justify your answer mathe-
matically.
3. The joint density for (X, Y) is given by
fxy (x, y) = 1/n²
x = 1, 2, 3, . ..,n
y = 1, 2, 3, . .. , n
(a) Verify that fxy (x, y) satisfies the conditions necessary to be a density.
(b) Find the marginal densities for X and Y.
(c) Are X and Y independent?
4. The joint density for (X, Y) is given by
fxy (x, y) = 2/n(n + 1)
1<y<x<n
n a positive integer
(a) Verify that fyy (x, y) satisfies the conditions necessary to be a density. Hir
The sum of the first n integers is given by n(n + 1) /2.d
(b) Find the marginal densities for X and Y. Hint: Draw a picture of the regi
over which (X, Y ) is defined.
(c) Are X and Y independent?
(d) Assume that n = 5. Use the joint density to find P[X< 3 and Y< 2]. Fï
P[X < 3] and P[Y < 2]. Hint: Draw a picture of the region over wh
(X, Y) is defined.
S. The two most common types of errors made by programmers are syntax err
and errors in logic. For a simple language such as BASIC the number of s
errors is usually small. Let X denote the number of syntax errors and Y
number of errors in logic made on the first run of a BASIC program. Assu
that the joint density for (X, Y) is as shown in Table 5.6.
3.
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