2. Let U be a continuous random variable with the following probability density function: g(t) =...
2. Let U be a continuous random variable with the following probability density function: 1+t g(t) = 1-t -1 < t < 0 0 <t<1 otherwise 0 a. Verify that g(t) is indeed a probability density function. [5] b. Compute the expected value, E(U), and variance, V(U), of U. (10)
Let U be a continuous random variable with the following probability density function: g(t) = 1+t -1<t< 0 1-t 0<t<1 0 otherwise a. Verify that g(t) is indeed a probability density function. [5] b. Compute the expected value, E(U), and variance, V(U), of U. (10)
2. Let U be a continuous random variable with the following probability density function: 1+1 -1 <t<o g(t) = { 1-1 03151 0 otherwise a. Verify that g(t) is indeed a probability density function. [5] b. Compute the expected value, E(U), and variance, V(U), of U. (10)
2te-t2 = { t> 0 6. Let g(t) be the probability density function of the continuous 0 t < 0 random variable X. a. Verify that g(t) is indeed a probability density function. [8] b. Find the median of X, i.e. the number m such that P(X 5 m) = į = 0.5. [7]
6. Let g(t) = { 2te** t 20 6. Let g(t) be the probability density function of the continuous 0 t<0 random variable X. a. Verify that g(t) is indeed a probability density function. [8] b. Find the median of X, i.e. the number m such that P(x = m) = { = 0.5. [7]
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].
2te-t2 { 2te-1 = t> 0 6. Let g(t) be the probability density function of the continuous 0 t < 0 random variable X. a. Verify that g(t) is indeed a probability density function. [8] b. Find the median of X, i.e. the number m such that P(X <m) = } = 0.5. [7]
10. Let X be a continuous random variable with probability density function -T xe h(x) = { x > 0 x < 0 0 a. Verify that h(x) is a valid probability density function. [7] b. Compute the expected value E(X) and variance V(X) of X. [8]
{ 2te-2 t> 0 6. Let g(t) be the probability density function of the continuous 0 t< 0 random variable X. a. Verify that g(t) is indeed a probability density function. [8] b. Find the median of X, i.e. the number m such that P(X <m) = } = 0.5. [7]
b. Let X be a continuous random variable with probability density function f(x) = kx2 if – 1 < x < 2 ) otherwise Find k, and then find P(|X| > 1/2).