Let X be a continuous random variable with the probability density ?(?) = 3?2 for values of x in [0,1], and ?(?) = 0 elsewhere. Compute the expected value and variance of X.
Let X be a continuous random variable with the probability density ?(?) = 3?2 for values...
10. Let X be a continuous random variable with probability density function -2 хе x > 0 h(z) = { { 0 x < 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]
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.
Q 2. The probability density function of the continuous random variable X is given by Shell, -<< 0. elsewhere. f(x) = {&e*, -40<3<20 (a) Derive the moment generating function of the continuous random variable X. (b) Use the moment generating function in (a) to find the mean and variance of X.
1. Let X be a continuous random variable with the probability density function fx(x) = 0 35x57, zero elsewhere. Let Y be a Uniform (3, 7) random variable. Suppose that X and Y are independent. Find the probability distribution of W = X+Y.
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)
5. Find the moment generating function of the continuous random variable X whose a. probability density is given by )-3 or 36 0 elsewhere find the values of μ and σ2. b, Let X have an exponential distribution with a mean of θ = 15 . Compute a. 6. P(10 < X <20); b. P(X>20), c. P(X>30X > 10), the variance and the moment generating function of x. d.
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]
4. Let X be a continuous random variable with probability density 1 0< x<3 -x + k =6 f(x) elsewhere 0, Evaluate k. a. b. Find P(1 < X< 2). c. Find E(X) d. Find e. Find ox. 4. Let X be a continuous random variable with probability density 1 0
2. 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+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)