Q21. Consider a linear amplifier defined by input-output relation Y = ax +b where a and b are constants. The input X is a Gaussian random variable with mean m and variance o. Determine the probability density function of the random variable Y at the amplifier output.

A box contains four 75W lightbulbs, three 60W lightbulbs, and three burned- out lightbulbs. Two bulbs are selected without replacement at random from the box. The bulbs are selected one at a time. Let X represents the number of 75W bulbs selected, and Y represents the number of 60W bulbs selected. i. Construct the joint probability mass function table of X and Y. ii. Find the mean of X.

Let x1  and x1 be independent standard normal random variables. If Y1= x1 + 2x2 and  Y2 = 2x1 - x2, find Fy1,y2(1,2)

A certain automobile manufacturer equips a particular model with either a turbo or a standard engine. Let X1 and X2 be fuel efficiencies for independently and randomly selected turbo and standard cars, respectively. Assume the two random variables are normally distributed with the following parameters: H = 22 and o,2 =1.44 (mean and variance of X1] Hi = 26 and = 2.25 [mean and variance of X2] Define a random variable Y as the difference between the first engine minus three times the second engine, X, - 3X, Find V(Y), the variance of the linear combination Y