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Random Variables for GATE
Question 1
Suppose you break a stick of unit length at a point chosen uniformly at random. Then the expected length of the shorter stick is ________
0.24 to 0.27
0.15 to 0.30
0.20 to 0.30
0.10 to 0.15
Question 2
Consider the two statements.
S1: There exist random variables X and Y such that
S2: For all random variables X and
Which one of the following choices is correct?
Both S1 and S2 are true
S1 is true, but S2 is false
S1 is false, but S2 is true
Both S1 and S2 are false
Question 3
Consider a finite sequence of random values X = { x1, x2,…, xn}. Let μx be the mean and σx be the standard deviation of X. Let another finite sequence Y of equal length be derived from this as yi = a*xi + b, where a and b are positive constants. Let μy be the mean and σy be the standard deviation of this sequence. Which one of the following statements is INCORRECT?
Index position of mode of X in X is the same as the index position of mode of Y in Y.
Index position of median of X in X is the same as the index position of median of Y in Y.
μy = aμx+b
σy = aσx+b
Question 4
Consider a company that assembles computers. The probability of a faulty assembly of any computer is p. The company therefore subjects each computer to a testing process.This testing process gives the correct result for any computer with a probability of q. What is the probability of a computer being declared faulty?
pq + (1 – p)(1 – q)
(1 – q) p
(1 – p) q
pq
Question 5
In a multi-user operating system on an average, 20 requests are made to use a particular resource per hour. The arrival of requests follows a Poisson distribution. The probability that either one, three or five requests are made in 45 minutes is given by :
6.9 × 106 × e-20
1.02 × 106 × e-20
6.9 × 103 × e-20
1.02 × 103 × e-20
Question 6
A point is randomly selected with uniform probability in the X-Y plane within the rectangle with corners at (0,0), (1,0), (1,2) and (0,2). If p is the length of the position vector of the point, the expected value of p2 is
2/3
1
4/3
5/3
Question 7
A program consists of two modules executed sequentially. Let f1(t) and f2(t) respectively denote the probability density functions of time taken to execute the two modules. The probability density function of the overall time taken to execute the program is given by :
Question 8
Suppose Y is distributed uniformly in the open interval (1, 6). The probability that the polynomial 3x2 + 6xY + 3Y + 6 has only real roots is (rounded off to 1 decimal place) _________.
option : 0.80
0.80
0.08
0.50
0.58
Question 9
If a random variable X has a Poisson distribution with mean 5, then the expression E[(X + 2)2] equals _____.
25
45
54
50
Question 10
A probability density function on the interval [a, 1] is given by 1 / x2 and outside this interval
the value of the function is zero. The value of a is :
0.05
0.075
0.25
0.5
There are 10 questions to complete.
