help me with these two probability questions.
help me with these two probability questions. 12. (3 points) Let f(x) = kx3 +2 -1...
Let X have probability density function f(2)= k(1+x) -3 for 0 < x < oo and f(x) = 0 elsewhere. a. Find the constant k and Find the c.d.f. of X. b. Find the expected value and the variance of X. Are both well defined? c. Suppose you are required to generate a random variable X with the probability density function f(x). You have available to you a computer program that will generate a random variable U having a U[0,...
1. (15 points) Let X be a continuous random variable with probability density function f (x) c(1-), 0 < 1, where c is a constant. i) Find the constant c ii) What is the distribution function of X? ii) Let Y 1x<0.5 Find the conditional expectation E(X|Y). 1. (15 points) Let X be a continuous random variable with probability density function f (x) c(1-), 0
1. 20 points Let X be a random variable with the following probability density function: f(x)--e+1" with ? > 0, ? > 0, constants x > ?, (a) 5 points Find the value of constant c that makes f(x) a valid probability mass function. (b) 5 points Find the cumulative distribution function (CDF) of X.
1. Let X be a continuous random variable with probability density function f(x) = { if x > 2 otherwise 0 Check that f(-x) is indeed a probability density function. Find P(X > 5) and E[X]. 2. Let X be a continuous random variable with probability density function f(x) = = { SE otherwise where c is a constant. Find c, and E[X].
[4+3+3 Points] 9. For the following probability density function, f(x) -k for 0 <x < 0.5 f(x) - 3x2 for 0.5 < x < 1 f(x) -0 otherwise What is the value of k? Find the median value of x Find the probability that X<0.75 a. b.
2. Let X and Y be continuous random variables with joint probability density function fx,y(x,y) 0, otherwise (a) Compute the value of k that will make f(x, y) a legitimate joint probability density function. Use f(x.y) with that value of k as the joint probability density function of X, Y in parts (b),(c).(d),(e (b) Find the probability density functions of X and Y. (c) Find the expected values of X, Y and XY (d) Compute the covariance Cov(X,Y) of X...
Verify Property 2 of the definition of a probability density function over the given interval. f(x)=3, [03] Next, determine F(x). First, find the antiderivative off. (3 dx = 3x 3x+C Let C = 0 in the expression obtained above and let the resulting expression be F(x). Evaluate the result over the far right side of the formula for theprea. 0-0 [0,1] using area =
i need help with both! QUESUU 41 POINTS Let Xhave the probability density function f(x) = 1/108, 0 < x < 108, so that X-Unif (0, 108). Calculate the median of X. Round answer to the nearest hundredths. Let Z be the standard normal distribution, Z-N(0,1). If the area between -z and zis 0.7776, find z.
Problem 3. The probability density function of X is f(x)o otherwise where k is some positive number. (a) Determine the value of the constant k. (b) Find the distribution function F(x)- P(X S r). (c) Compute μ-E(X), the mean of X. (d) Compute σ2-Var(X), the variance of X.
3. (10 points) Let X be continuous random variable with probability density function: fx(x) = 7x2 for 1<<2 Compute the expectation and variance of X 4. (10 points) Let X be a discrete random variable uniformly distributed on the integers 1.... , n and Y on the integers 1,...,m. Where 0 < n S m are integers. Assume X and Y are independent. Compute the probability X-Y. Compute E[x-Y.