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

Q: Calculate E((X +Y)²).

A:

Q: Two random variables X andY are independent if f(x, y) = f1(x). f2(y) Select one: O True O False

A:

Q: Suppose X is a discrete random variable with finite first and second moments. Show that E (x…

A: It is given that X is a discrete random variable with finite first and second moment.

Q: If xy Xy .... Xy is a random sample from the distribution. f (x, 0) er (1- 0)-x, x = 0, 1 and 0< 0<1…

A: Answer: For the given data,

Q: (34) Let X be a random variable with p.d.f. 1<x<» k f(x) = { find k. O.w

A: Formula used: fx=x-2k           1&lt;x&lt;∞0                   Otherwise

Q: ii) Let X be a random variable taking three values: P(X= a₁) = P₁, P(X=a₂) = P2, P(X=a3) = P3, where…

A: Given Information: Let X be a random variable taking three values: PX=a1=p1, PX=a2=p2, PX=a3=p3,…

Q: If X is an exponential random variable with parameter 1, and c > 0, show cX is expor

A:

Q: Let X1, X2,... be independent Exponential (3) random variables and Z = } E=1 X;. If Mz (t) is the…

A:

Q: E(Y) –> P(Y > n).

A:

Q: O Find E[S„] and Var[S„].

A: It is given that Xis are iid and follows Uniform distribution with parameter a and b.

Q: 3. If X and Y are independent uniform (0, 1) random variables, show that 2 for a > 0 (a + 1)(a + 2)…

A: X and Y are independent U(0,1) random variables:Objective: Show that:

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: 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)&gt;= (E(X))2 when we find Variance.

Q: Let X be a non-negative random variable that is not a constant. The product E(1/X) times E(X) is…

A: The question is asking whether the product of the expected value of a non-negative random variable X…

Q: uppose X and Y are independent random variables. If the pdf of x is xe ",x ≥ 0 and the pdf of y is…

A: Solution

Q: (1) E(E(X)) = E(X) (2) E(X – E(X)) = 0 (3) Var(X) = E(X²) – [E(X)]² (4) Given constants a and b,…

A: Discrete random variable is defined as a variable which can take only specific values. For example…

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 fx,y (x, y) be the joint probability function for the random vector (X,Y) (discrete or…

A: fx(1) = 0 f(1,y) = 0

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: prove that var (XY) == 0} 03+ 2} · o3+ 13.03

A:

Q: 3. Let X and Y be two independent Bernoulli (0.5) random variables. Let Z = X + Y. Show that Var (Z)…

A: Bernoulli Distribution: You should understand what a Bernoulli random variable is and know its…

Q: what is E[XY]?

A:

Q: -. Determine the constant c so that f(x)=cx, x=1, 2, 3,...,10 satisfies the conditions of being a p.…

A: The given probability distribution can be desctibed as X 1 2 3 4 5 6 7 8 9 10 f(x) c 2c 3c 4c…

Q: Let X1, X2, X3 form a random sample from U(0,1). Find a- The distribution of the second order…

A:

Q: If Xn, n > 1, independent random variables and Xn 4 X (a random variable X), are

A: A degenerate distribution (sometimes called a constant distribution) is a distribution of a…

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

A:

Q: f X1 and X2 are independent exponential random variables with respective parameters λ1 and λ2, find…

A: Given:    X1, X2 are two independent exponential random variables with respective parameters λ1 and…

Q: 2. If X is a random variable taking values 0,1,2,.... and P(X) = ab*, where a and b are positive…

A: i) First part:ii) second part:

Q: 8. IFX is a continuous random variable and y = ax + b then the expected valued of y = ? a. aE(x) b.…

A: y=ax+b

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

A:

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

A:

Q: If x, x2 ... Xn is a random sample from N (µ, o²) popula- tion, find sfficient estimator for µ and…

A: ANSWER: For the given data,

Q: Let X be a random variable with probabilities as shown in Table 3.7. Table 3.7 Values of X and P(x)…

A:

Q: = 4 and V(X) = 1, which of the following statements Let X be a continuous random variable that only…

A: Given mean=mu=4, variance=sigma^2=1 We solve by using chebyshev's inequality…

Q: (34) Let X be a random variable with p.d.f. x= 1,2,3 f(x) ={ , find (1) k (2) E(²) O.w

A: X is a random variable with pdf: The sum is always 1. So,

Q: If U1, ..., Un are independent uniform random variables, find E(U(n) −U(1)).

A: U1, U2, ....., Un are independent uniform random variables in the interval [a,b]. The density of Ui…

Q: Suppose that W is a random variable with E(W4)&lt; ∞. Show thatE(W2) &lt; ∞.

A: Given: Jensen’s inequality says that if g(.) is a convex function:

Q: Suppose that X₁, X2, . . ., X₁ and Y₁, Y₂ , . . Yn are independent . random samples from populations…

A: Let X=X1,X2,..,Xn and Y=Y1,Y2,...,Yn are independent random samples from a population.Here,  μ1 and…

Q: B5. Let X₁, X2,..., Xn be IID random variable with common expectation and common variance o², and…

A: Let X1, X2, . . . ., Xn be i.i.d. random variable with a common expectation E(Xi)=μ and common…