Consider the joint PDF for the type of customer service X (0 = telephonic hotline, 1 = Email) and of satisfaction score Y (1 = unsatisfied, 2 = satisfied, 3 = very satisfied)

State whether the following statements are True or False and also give the reason in support of your answer. (5×2=10)

(a) Sample space of a (i) random experiment tossing two coins simultaneously and (ii) One coin two times is the same.

(b) Standard deviation of a random variable X may take any real value, i.e. its value lies in the interval.[- ∞ , ∞ ]

A group of 100 people are comparing their birthdays (as usual, assume their birthdays are independent and not on February 29, etc.). Find the expected number of pairs of people with the same birthday, and the expected number of days in the year on which at least two of these people were born.

A random 5-card poker hand is dealt from a standard deck of cards. Find the probability (in terms of binomial coefficients) of getting a flush (all 5 cards being of the same suit: do not count a royal flush, which is a flush with an ace, king, queen, jack and 10).

State whether the following statements are True or False. Give reason in support of your answer: (5×2=10)

(a) If the probability of non rejection of H₀ when H₁ is true is 0.4 then power of the test will be 0.6.

(b) If T₁ and T₂ are two estimators of the parameter θ such that Var(T₁) = 1/n and Var(T₂) = n then T₁ is more efficient than T₂.

(c) A 95% confidence interval is smaller than 99% confidence interval.

(d) If the level of significance is the same, the area of the rejection region in a two-tailed test is less than that in a one-tailed test.

(e) Non parametric tests are more powerful than the parametric tests.

State Monty Hall problem and solve it.

There are 4 black, 3 blue and 8 red balls in an urn. Three balls are selected one by one without replacement. What is the probability that:

(i) First ball drawn is black, second one is red and third one is blue.

(ii) All the three balls are of the same colour.