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 randomly

selected from a bin containing seven such gauges. Unknown to the
scientist, three of

the seven gauges giveimproper temperature readings. Let
X denote the number of

defectiye gauges selected and Y the number of
nondefective 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 See.
3.7 to verify the values

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 mathemati-

cally.