Probability Theory
Q11.
Consider Guwahati (G) and Delhi (D) whose temperatures can be classified as high (G), medium (M) and low (L). Let P(H_{G}) denote the probability that Guwahati has high temperature. Similarly, P(M_{G}) and P(L_{G} ) denotes the probability of Guwahati having medium and low temperatures respectively. Similarly, we use P(H_{D}),P(M_{D}) and P(L_{D}) for Delhi. The following table gives the conditional probabilities for Delhi's temperature given Guwahati's temperature. Consider the first row in the table above. The first entry denotes that if Guwahati has high temperature (H_{G}) then the probability of Delhi also having a high temperature (H_{D}) is 0.40; i.e., (H_{D}|H_{G}) = 0.40. Similarly, the next two entries are P(M_{D}| H_{G})= 0.48 and P(L_{D}|H_{G}) = 0.12. Similarly for the other rows. If it is known that P(H_{G})= 0.2, P(M_{G})= 0.5, andP(L_{G})= 0.3, then the probability (correct to two decimal places) that Guwahati has high temperature given that Delhi has high temperature is _______Q12.
For a given biased coin, the probability that the outcome of a toss is a head is 0.4. This coin is tossed 1,000 times. Let X denote the random variable whose value is the number of times that head appeared in these 1,000 tosses. The standard deviation of X (rounded to 2 decimal place) is ________Q13.
Two people, P and Q, decide to independently roll two identical dice, each with 6 faces, numbered 1 to 6. The person with the lower number wins. In case of a tie, they roll the dice repeatedly until there is no tie. Define a trial as a throw of the dice by P and Q. Assume that all 6 numbers on each dice are equi-probable and that all trials are independent. The probability (rounded to 3 decimal places) that one of them wins on the third trial isQ14.
For any discrete random variable X, with probability mass function P(X=j)=p_{j}, p_{j}\geq 0, j \in \{0,....N\} and \sum_{j=0}^{N}p_{j}=1, define the polynomial function g_{x}(z)=\sum_{j=0}^{N}p_{j}z^{j}. For a certain discrete random variable Y, there exists a scalar \beta \in [0,1] such that g_{Y}(z)=(1-\beta +\beta z)^{N}. The expectation of Y isQ15.
Let A and B be any two arbitrary events, then, which one of the following is TRUE?Q16.
Suppose X_{i} for i =1,2,3 are independent and identically distributed random variables whose probability mass functions are Pr[X_{i}=0]=Pr[X_{i}=1]=1/2 for i=1,2,3. Define another random variable Y=X_{1}X_{2}\oplus X_{3}, \; where \; \oplus denotes XOR. Then Pr[Y=0|X_{3}=0]=_______________.Q17.
A probability density function on the interval [a,1] is given by 1/x^{2} and outside this interval the value of the function is zero.The value of a is __________.Q18.
Each of the nine words in the sentence "The quick brown fox jumps over the lazy og" is written on a separate piece of paper. These nine pieces of paper are kept in a box. One of the pieces is drawn at random from the box. The expected length of the word drawn is _____________. (The answer should be rounded to one decimal place)Q19.
Let S be a sample space and two mutually exclusive events A and B be such that A\cupB = S. If P(.) denotes the probability of the event, the maximum value of P(A)P(B) is ______