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?

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

The value of  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        

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

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___________.

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

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