2. Let X be a continuous random variable with pdf f(x) = { cr", [w] <1,...
2. Let X be a continuous random variable with pdf ( cx?, [xl < 1, f(x) = { 10, otherwise, where the parameter c is constant (with respect to x). (a) Find the constant c. (b) Compute the cumulative distribution function F(x) of X. (c) Use F(x) (from b) to determine P(X > 1/2). (d) Find E(X) and V(X).
2. Let X be a continuous random variable with pdf ( cx?, |a| 51, f(x) = { 10, otherwise, where the parameter c is constant (with respect to x). (a) Find the constant c. (b) Compute the cumulative distribution function F(x) of X. (c) Use F(x) (from b) to determine P(X > 1/2). (d) Find E(X) and V(X).
2. Let X be a continuous random variable with pdf ca2, 1 f(x) otherwise, where the parameter c is constant (with respect to x) (a) Find the constant c (b) Compute the cumulative distribution function F(x) of X (c) Use F(x) (from b) to determine P(X 1/2) (d) Find E(X) and V(X)
LI CONTINUOUS DIST Let X be a random variable with pdf -cx, -2<x<0 f(x)={cx, 0<x<2 otherwise where c is a constant. a. Find the value of c. b. Find the mean of X. C. Find the variance of X. d. Find P(-1 < X < 2). e. Find P(X>1/2). f. Find the third quartile.
8. Let X and Y be a random variable with joint continuous pdf: f(x,y)- 0< y <1 0, otherwise a. b. c. Find the marginal PDF of X and Y Find the E(X) and Var(X) Find the P(X> Y)
2. Suppose X is a continuous random variable with the probability density function (i.e., pdf) given by f(x) - 3x2; 0< x < 1, - 0; otherwise Find the cumulative distribution function (i.e., cdf) of Y = X3 first and then use it to find the pdf of Y, E(Y) and V(Y)
3. Let X be a continuous random variable with the following PDF f(x) = ( ke 2 x 20 f(x)= otherwise where k is a positive constant. (a). Find the value of k. (b). Find the 90th percentile 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].
Let X be a continuous random variable with the following PDF 6x(1 – x) if 0 < x < 1 fx(x) = 3 0.w. Suppose that we know Y | X = x ~ Geometric(x). Find the posterior density of X given Y = 2, i.e., fxY (2|2).
3. Consider a continuous random variable X with pdf given by 0, otherwise This is called the exponential distribution with parameter X. (a) Sketch the pdf and show that this is a true pdf by verifying that it integrates to 1 (b) Find P(X < 1) for λ (c) Find P(X > 1.7) for λ : 1