Let X and Y be two real-valued random variables with

, and , where E denotes the expectation operator.

The value of  that minimizes   is

(Round off to one decimal place)

Consider discrete random variables  and  with probabilities as follows:

Given , the expected value of  is

Let  and  be continuous random variables with probability density functions  and , respectively. Further, let  and

Which one of the following options is correct?

Let  be a discrete random variable that is uniformly distributed over the set . Which of the following random variables is/are uniformly distributed?

One million random numbers are generated from a statistically stationary process with a Gaussian distribution with mean zero and standard deviation .

The  is estimated by randomly drawing out 10,000 numbers of samples . The estimates  are computed in the following two ways.

Three points in the -y plane are  and . The value of the slope of the best fit straight line in the least square sense is __________ (Round off to 2 decimal places).

The expected number of trials for first occurrence of a "head" in a biased coin is known to be 4. The probability of first occurrence of a "head" in the second trial is __________ (Round off to 3 decimal places).

Let the probability density function of a random variable  be given as

The value of  is  __________.

Suppose the probability that a coin toss shows "head" is , where . The coin is tossed repeatedly until the first "head" appears. The expected number of tosses required is

The mean-square of a zero-mean random process is  where k is Boltzmann’s constant, T is the absolute temperature, and C is a capacitance. The standard deviation of the random process is        

The probability of a resistor being defective is 0.02. there are 50 such resistors in a circuit. The probability of two or more defective resistors in the circuit (round off to two decimal places) is ________.

An urn contains 5 red balls and 5 black balls. In the first draw, one ball is picked at random and discarded without noticing its color. The probability to get a red ball in the second draw is

Assume that in a traffic junction, the cycle of the traffic signal lights is 2 minutes of green (vehicle does not stop) and 3 minutes of red (vehicle stops). Consider that the arrival time of vehicles at the junction is uniformly distributed over 5 minute cycle. The expected waiting time (in minutes) for the vehicle at the junction is __________.

A person decides to toss a fair coin repeatedly until he gets a head. He will make at most 3 tosses. Let the random variable Y denote the number of heads. The value of var {Y}, where var denotes the variance, equals

Candidates were asked to come to an interview with 3 pens each. Black, blue, green and red were the permitted pen colors that the candidates could bring. The probability that a candidates comes with all 3 pens having the same colour is _______.

Let the probability density function of a random variable, X, be given as:

where u(x) is the unit step function.

Then the value of ‘a’ and , respectively, are

A random variable X has probability density function f(x) as given below:

If the expected value , then Pr[X < 0.5] is___________.

Two players, A and B, alternately keep rolling a fair dice. The person to get a six first wins the game. Given that player A starts the game, the probability that A wins the game is

Two coins R and S are tossed. The 4 joint events  have probabilities 0.28, 0.18, 0.30, 0.24, respectively, where H represents head and T represents tail. Which one of the following is TRUE?

A fair coin is tossed n times. The probability that the difference between the number of heads and tails is (n – 3) is

Consider a dice with the property that the probability of a face with n dots showing up is proportional to n. The probability of the face with three dots showing up is _______.

Let x be a random variable with probability density function

The probability  is ____________.

Lifetime of an electric bulb is a random variable with density, where x is measured in years. If the minimum and maximum lifetimes of bulb are 1 and 2 years respectively, then the value of k is ___________.

A continuous random variable  has a probability density function. Then P is

Two independent random variables X and Y are uniformly distributed in the interval [-1, 1]. The probability that max [X, Y] is less than ½ is

A fair coin is tossed till a head appears for the first time. The probability that the number of required tosses is odd, is

A box contains 4 white balls and 3 red balls, In succession, two balls are randomly selected and removed from the box. Given that the first removed ball is white, the probability that the second removed ball is red is

Assume for simplicity that N people, all born in April (a month of 30 days), are collected in a room.  Consider the event of at least two people in the room being born on the same date of the month, even if in different years. e.g. 1980 and 1985. What is the smallest N so that the probability of this event exceeds 0.5 ?        

X is a uniformly distributed random variable that takes values between 0 and 1. The value of  will be

A loaded dice has following probability distribution of occurrences

If three identical dice as the above are thrown, the probability of occurrence of values

1, 5 and 6 on the three dice is

Two fair dice are rolled and the sum r of the numbers turned is considered.

If P and Q are two random events, then the following is TRUE:

A fair coin is tossed three times in succession. If the first toss produces a head, then the probability of getting exactly two heads in three tosses is: