Question: S Let X be a real-valued random variable on a probability space . ;F; P/. Show that X is independent of itself if and only

S Let X be a real-valued random variable on a probability space . ;F; P/. Show that X is independent of itself if and only if X is constant with probability 1, i.e., if there exists a constant c 2 R such that P.X D c/ D 1. Hint: Consider the distribution function of X.

Step by Step Solution

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock blur-text-image

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Students Have Also Explored These Related Probability Statistics Questions!

2.3 Functions of discrete random variables Let X be a discrete random variable on the probability space (,F , P) and let g : R R. It is easy to check that Y = g(X) is a discrete random variable on...

Markov chain. 8 Consider an irreducible Markov chain on a finite state space, such that the transition matrix is symmetric (pij = pji for all i, j ? S). Find the stationary distribution. 9 Markov...

Let be a sample space, and let a, b be two distinct outcomes. Consider the -algebra A = {, {a}, {a} c , } and a probability measure P : A [0, 1]. Let X be a real random variable on the probability...

4Let be a sample space, and let a, b be two distinct outcomes. Consider the -algebra A = {, {a}, {a} c , } and a probability measure P: A [0, 1]. Let X be a real random variable on the probability...

Let X 0 a.s. be a random variable on some probability space (, F, P ). Let : [0, ) R be absolutely continuous i.e., (x) = x 0 (t) dt. Using the approach we used in class (via Fubini's theorem) show...

Exercise 2.8 If X and Y are discrete random variables on the probability space (,F , P), show that U and V are discrete random variables on this space also, where U() = X() + Y (), V() = X()Y(), for...

Question No. 17 17- let X be a discrete random variable on a probability space [2, 37, P ] given by the following table: x, values of X -1 3 6 P. = P(X = =) 0.500 0.333 0.167 Which of the following...

Moment generating function. Let $X$ be a positive random variable on the probability space $(\Omega, \mathscr{A}, \mathbb{P})$. The function $\phi_{X}(t):=\int e^{-t X} d \mathbb{P}$ is called...

A first course in probability 10th ch. 4 p195 4.10Help me .. 4.10. Let X be a binomial random variable with parame- ters n and p. Show that: E[(1 - p) * ] = (1 - p2)n

5. Let Y be an integrable random variable on a probability space (0, F, P) and let G be a sub-o-algebra of F. Based on the information in 9, we can form the estimate E[Y G] of Y and define the error...

Consider the scenario where a child buys kids meals at a local restaurant in order to complete his collection of superhero action figures. Recall the states were X [k] { 0, 1, 2, 3, 4} where X [k]...

The state lottery's million-dollar payout provides for $1million to be paid over 19 years in 20 payments of $50,000. The first $50,000 payment is made immediately, and the 19 remaining $50,000...

Does the Assistant Controller's failure to adequately review the Vendor Change Form represent a deficiency in the design or operating

According to the Highway Loss Data Institute (HLDI), "Hybrid [automobiles] have a safety edge over their conventional twins when it comes to shielding their occupants from injuries in crashes" (HLDI...