B) Consider a continuous random variable X with the probability density function (PDF): ce , 1

10) Let f(x) by the probability density function of some continuous random variable. Explain why it must be true that f(x) dx =1.

Q1) CDF, PDF, Expectation and Variance The cumulative distribution function (CDF) of random variable Y is y 1. a) Find the probability density function, fy(y), of random variable Y. b) Show that f fy (y)dy = 1. c) What is E[Y]? d) What is VAR[Y]?

part d only

Let (X,Y) be bivariate random variables having joint probability density function as x) = ;(x + y) 0sx< 2,0 < y s 2 otherwise Find the following: (i) The correlation coefficient between X and Y. (ii)E (X"Y$)

The life lengths of two transistors in an electronic circuit is a random vector (X; Y) where X is the life length of transistor 1 and Y is the life length of transistor 2. The joint probability density function of (X; Y) is given by x 2 0, y 2 0 fx.,fx.v) = 20 else Then the probability that the first transistor burned during half hour given that the second one lasts at least half hour equals Select one: a. 0.606 b. 0.3935 C. 0.6318 d. 0.3669 e. 0.7772

Let X and Y are two random variables with p.d.f as F(x,y) = 3x^2y+3xy^2,0 < x< 1,0 < y < 1. (i) Find conditional density of X given Y. (ii) Find the mean, mode and standard deviation of r.v Y. (iii) Correlation between X and Y. (iv) Check that X and Y are independent or not?

6. Let Yi and Y2 be independent exponential random variables, both with mean 0>0. Let U = - and V = Y, +Y,. Using a change of variable technique, %3D Y, +Y; (a) find the probability density function of U. (b) find the probability density function of V.

Let the probability density function of the random variable X is x f(x) Then Mx (t) is 0 , x = 1,2,3,... , otherwise

1) Let x be a uniform random variable in the interval (0, 1). Calculate the density function of probability of the random variable y where y = − ln x.