Let X be a discrete random variable such that P(X = n) = (1 − p)"−¹p. Prove that E(X-¹) = -pln(p) 1-p

Q: 4. (a) Consider a nonnegative, integer valued random variable Y, and a random sample X1, ·.. , Xn.…

A: Expectation gives the most likely value of a random variable.

Q: Find fQ|X (q|x) for x ∈ {0, 1} and all q.

A: It is a fact of statistics. It is widely used.

Q: tes, sno W inut E [XY] = E[X] E[Y] E [g1(X) g2(X)] = E [g1 (X)] E [g2 (X)1 and

A:

Q: If E and F are two independent events, such that P (E ∩ F ) = 1/6 , P (E' ∩ F' ) = 1/3 and (P (E) −…

A:

Q: 3. Let the random variable X have the pmf f(x) = = a) E(X) b) E(X²) c) E(3X²2X + 4) (x+1)² for x =…

A: The provided information is as follows:The probability mass function of  is .

Q: Let X be a random variable such that E(X) = 3 and E(X2) = 13. Use the Chebyshev inequality to…

A:

Q: If X be a continuous random variable with be -bx if x >0 f(x)= otherwise then the moment generating…

A: The random variables can be categorized into 2, continuous random variable and discrete random…

Q: Let M and N be two independent events such that P(M) = 1/3 and P(N) = 1/4. Find P(M/N')

A:

Q: 4. Let X be a random variable taking positive values and assume that E(X) exists. Which of the…

A: Option (I) is correct as E(X2)>= (E(X))2 when we find Variance.

Q: (40) Let X be a random variable with p.d.f. (k +4)e¯3x f(x) ={ 01) O.w

A: a)Use the condition of valid pdf to find the value of k…

Q: Let X be a random variable. Find E(Y) where 3 Y = (X – E(X)) (a) 1 (d) Q (b) 3/4 (e) -1 (с) —3/4 -

A:

Q: Suppose the random variable y is a function of several independent random variables, say…

A: Given that the random variable y is a function of several independent random variables, say…

Q: (33) Let X be a random variable with p.d.f. e2k x= 1,2,3 f(x)=- find (1) k (2) E(x+1) O.W

A: Given that X is a random variable with p.d.f f(x)=e2k ; x=1,2,30     ; Otherwise.

Q: (13) Let X be a random variable with p.d.f. 2e-2x 0<x<0 f(x) ={ , find E(e2*) O.W

A: Solution :  Given :  f(x) = 2e-2x    0&lt;x&lt;∞0   o.w     E( e-2x )  = ∫x e-2xf(x) dx…

Q: ). Let M = ln(5*X) be a derived random variable. What is E[M]?

A:

Q: 4. If the random variable X has an show that for a > 0, b>0, exponential distrib f(x) = x/4, for x ≥…

A: For the random variable X~Exponential1λ, it is given that: fx=1λe-xλ, for x≥0

Q: Derive the MGF for an exponential random variable X ~ exp(A) (try the Kernel trick!) O eta t-入…

A:

Q: Fact: If X is a compound random variable with X = ₁X₁, ther E(X) = E(X₁) E(N) a. when X are all…

A:

Q: Let X be a random variable with pdf given by fX(x) = 1/[π(1 + x2)] for all real number x. Prove that…

A: Solution

Q: Let X be a positive random variable; i.e P(X ≤ 0) = 0. Which one is correct? (1) E (+) (²) (11)…

A: Given:X is a positive random variable, meaning .We need to determine which of the statements is…

Q: Consider the time series model x, = 0, +(1-0,B)z,, 2, - NID (0,02) Derive: b) the variance of x,. c)…

A: Given the time series model : xt=θ0+1-θ1Bzt ,  zt~NID0, σz2

Q: Suppose that the waiting time X (in seconds) for the pedestrian signal at a particular street…

A: The random variable   represents the waiting time for the pedestrian signal. The PDF of random…

Q: Let X = (X1, X2)' be a random sample from Po(A), A > 0. Show using the definition %3D chat S(X) =…

A:

Q: 7. A discrete random variable X taking non-negative integer values has probability gen- erating…

A:

Q: If X1 and X2 are independent random variables with distribution given by P[Xi] = −1 = P[Xi = 1] =…

A: Construct the PMF of X1X2, So, P(X1X2 = x) = P(X1=1, X2 = 1) +P(X1=-1, X2 = -1)    if x =1P(X1=-1,…

Q: Suppose that a random variable X is an Exponential Random Variable with parameter β = 3. (a) What is…

A: Solution: From the given information, X follows an exponential distribution with parameter β=3.

Q: et X be a random variable which takes values in the interval ow that E(X)- a PIXI> a)

A: We are given,  X=Random variable which has the values between [-M, M] only. Hence, we have the…

Q: Suppose that X is a random variable for which E(X) = μ and Var(X) = σ2. Show that E[X(X − 1)] = μ(μ…

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Q: Let X1, …, Xn be independent lognormal random variables and let a1, …, an be constants. Show that…

A:

Q: (52) Let X be a random variable with p.d.f. 1 0<x<0 f(x) = ,find E(e") O.w -in

A:

Q: (40) Let X be a random variable with p.d.f. (k +4)e-3* f(x) =- 01) O.w

A:

Q: (4) Let X be a random variable with p.d.f. 2e-2x 0<x<0 f(x)= , find E(e2x) O.w

A: Random Variable: A random variable is a real valued function that assign a real number to each…

Q: (1) Let X be a random variable with p.d.f. e 0<x<0 k ƒ(x) = < find k. O.W

A: Let X be a random variable with pdf  fx=1ke-8x        0&lt;x&lt;∞0                         otherwise…

Q: Suppose that Y is an exponential random variable with λ = 4. Find. P[Y > (E(Y) + 2√√Var(Y))].

A: From given data we have : λ=4 and Y~exp(λ)