We say that a random variable X is symmetric if pX(n) = pX(−n) for all values of n. More generally, X is symmetric around c if pX(n) = pX(c − (n − c)). Prove that if X is symmetric around c, then E[X] = c.
We say that a random variable X is symmetric if pX(n) = pX(−n) for all values...
3 (17') The random variable X obeys the distribution Binomial(n,p) with n=3, p=0.4. (a) Write Px(x), the PMF of X. Be sure to write the value of Px(x) for all x from - to too. (b) Sketch the graph of the PMF Px [2] (c) Find E[X], the expected value of X. (d) Find Var[X], the variance of X.
Problem 4. Let A, B e Rmxn. We say that A is equivalent to B if there exist an invertible m x m n x n matrix Q such that PAQ = B. matrix P and an invertible (a) Prove that the relation "A is equivalent to B" is reflexive, symmetric, and transitive; i.e., prove that: (i) for all A E Rmx", A is equivalent to A; (ii) for all A, B e Rmxn, if A is equivalent to B...
2) Consider a random variable with the following probability distribution: P(X-0)-0., Px-1)-0.2, PX-2)-0.3, PX-3) -0.3, and PX-4)-0.1 A. Generate 400 values of this random variable with the given probability distribution using simulation. B. Compare the distribution of simulated values to the given probability distribution. Is the simulated distribution indicative of the given probability distribution? Explain why or why not. C. Compute the mean and standard deviation of the distribution of simulated values. How do these summary measures compare to the...
The probability mass function of a random variable X is given by PX(n) = (c)(λ^n) / n! , n = 0, 1, 2, . . . (a) Find c (Hint: use the relationship that (SUM) n=0->∞ ( x^n / n! = e^x ) (b) Now assume λ = 2, find P(X = 0) (c) Find P(X > 3)
The probability mass function of a random variable X is given by Px(n)r n- (a) Find c (Hint: use the relationship that Ση_0 n-e) (b) Now assume λ = 2, find P(X = 0) (c) Find P(X>3)
Part c please
Suppose that Zn n/2 where Zn is any random variable with Eena, say, with c0 and a E R fixed, and X is any other random variable. (a) Let > 0. Use Chebyshev's inequality to show that (b) For what values of o does the argument in part (a) prove that X converges in probability to X? (c) For the values of α identified in part (b), what other mode of convergence of X, to X is...
Suppose that X is a continuous random variable with density
pX(x) = ( Cx(1 − x) if x ∈ [0, 1] 0 if x < 0 or x > 1.
(a) Find C so that pX is a probability density function.
(b) Find the cumulative distribution of X.
(c) Calculate the probability that X ∈ (0.1, 0.9).
(d) Calculate the mean and the variance of X.
9.) Suppose that X is a continuous random variable with density C(1x) if E...
Exercise 3. We say that a family of random variables (X)teo.) converges in probability to a random variable X (notation: Xt-, X) if for everye> 0, liml+xPfXt-X) > e) = 0. Suppose that (Nt)t20 is a Poisson process of rate X. Show that Nt/t P+ λ. This shows that the rate measures the average frequency or density of arrivals. What about N1 λ?
Exercise 3. We say that a family of random variables (X)teo.) converges in probability to a random...
1. Let X be a random variable with variance ? > 0 and fx as a probability density function (pdf). The pdf is positive for all real numbers, that is fx(x) > 0. for all r ER Furthermore, the pdf fx is symmetric around zero, that is fx(x) = fx(-1), for all r ER Let y be the random variable given by Y = 4X2 +6X + with a,b,c E R. (i) For which values of a, b, and care...
Problem 3. Let X be a discrete random variable that takes values in N. Show that if X is memory-free then it must be the case that X Geo(p) for some p. (Hint a useful first step might be to show that P(X > t)= P(X > 1)' for all t E N.)
Problem 3. Let X be a discrete random variable that takes values in N. Show that if X is memory-free then it must be the case that...