2. Let Y = X^2+ X+1. (a) Evaluate the mean and variance of Y, if
X...
2. Let Y = X^2+ X+1. (a) Evaluate the mean and variance of Y, if
X is an exponential random variable. (b) Evaluate the mean and
variance of Y, if X is a Gaussian random variable
Let X have mean ? and variance ?^2. (a) What are the mean and
variance of -X? Explain intuitively. (b) Find constants a and b
such that the random variable Y=aX +b has mean 0 and variance 1
(this is called standardization).
1. Let X be a random variable with mean μ and variance σ . For a
∈ R, consider the expectation E ((X − a)2)
a) Write E((X −a)2) in terms of a,μ and σ2
b) For which value a is E ((X − a)2) minimal?
c) For the value a from part (b), what is E ((X − a)2)?
2. Suppose I have a group containing the following first- and
second-year university students from various countries. The first 3...
a)
Let y be the solution of the equation y ′ =
(y/x)+1+(y^2/x^2) satisfying the condition y ( 1 ) = 0.
Find the value of the function f ( x ) = (y ( x ))/x
at x = e^(pi/4) .
b)
Let y be the solution of the equation y ′ = (y/x) −
(y^2/x^2) satisfying the condition y ( 1 ) = 1. Find the
value of the function f ( x ) = x/(y(x))
at x = e .
c)
Let y be the solution...
2. Let X be a normal random variance with media 1 and variance
4. Consider a new variance A random variable T defined below:
T = -1 if X < -2
T = 0 if - 2 ≤ X ≤ 0
T = 1 if x>0
Find the moment generating function of T and, from it, calculate E (T) and Var (T).
2. Let X be a normal random variance with media 1 and variance
4. Consider a new variance A random variable T defined below: T =
-1 if X < -2 T = 0 if - 2 ≤ X ≤ 0 T = 1 if x>0 Find the
moment generating function of T and, from it, calculate E (T) and
Var (T).