Problem(3) (6 points) Consider the random variable X whose density is given by p(z) - ksin(x)...
Problem 6. Consider a random variable X whose cumulative distribution function (cdf) is given by 0 0.1 0.4 0.5 0.5 + q if -2 f 0 r< 2.2 if 2.2<a<3 If 3 < x < 4 We are also told that P(X > 3) = 0.1. (a) What is q? (b) Compute P(X2 -2> 2) (c) What is p(0)? What is p(1)? What is p(P(X S0)? (Here, p(.) denotes the probability mass function (pmf) for X) (d) Sketch a plot...
2) Consider a random variable Z with a uniform probability
density function given as UZ(-1,0) and X=4Z+4. a) Find and plot the
probability density function ( ) Xf x . b) Find and plot the
probability distribution function ( ) F x X . c) Find E[Z]. d) Find
E[X]. e) Find the correlation of Z and X. i. Are they correlated?
ii. Are they independent? Why?
2) Consider a random variable Z with a uniform probability density function given...
3. Consider two random variables X and Y, whose joint density function is given as follows. Let T be the triangle with vertices (0,0), (2,0), and (0,1). Then if (x, y for some constant K (a) (2 pts.) Find the constant K (b) (4 pts.) Find P(X +Y< 1) and P(X > Y). (c) (4 pts.) Find the marginal densities fx and fy. Conclude that X and Y are not independent
Consider the random variable X whose probability density function is given by k/x3 if x>r fx(x) = otherwise Suppose that r=5.2. Find the value of k that makes fx(x) a valid probability density function.
1. [4 points] Consider a random variable X whose probability distribution function is given by 0.4 ifx=0 0.3 if3 0 elsewhere (I) find the value of k (2) find the mean of X (3) find the variance of X
Suppose density function positively valued continuous random variable X has the probability a fx(x)kexp 20 fixed 0> 0 for 0 o0, some k > 0 and for (a) Find k such that f(x) satisfies the conditions for a probability density function (4 marks) (b) Derive expressions for E[X] and Var[X (c) Express the cumulative distribution function Fx(r) in terms of P(), the stan dard Normal cumulative distribution function (8 marks) (8 marks) (al) Derive the probability density function of Y...
5. (20%) Let X be a continuous random variable whose probability density function is fr(x) (a +bx)%0(x) (a) If Ex)f find a and b. (b) Give the cumulative distribution function F,(x) f()dt of X and Var(X) (c) Let A be any Borel set of R. Define P by P(A) [,f dm
5. (20%) Let X be a continuous random variable whose probability density function is fr(x) (a +bx)%0(x) (a) If Ex)f find a and b. (b) Give the cumulative distribution...
3. Let X be random variable with probability density function x(x)4 for 0 x 1, (Note: fx (x) = 0 outside this domain.) (a) Find E[X] and Var[X] (b) Let Y- X2 +5. Find E[Y] and Var[Y]. (c) Find PX 112 ).
3. A random variable X has the probability mass function P(x = k) = (a > 0, k =0,1,2...). (1 + a)! Find E[X], Var(X), and the Moment generating function My(t) = E[ex]
Problem 5. The joint density of X and Y is given by e" (z+y) fx.-otherwise. İf 0 < x < oo, 0 < y < 00, Consider the random variable Z-; a) Find the cumulative distribution function of Z b) What is the probability density function of Z?