. If there are n random variables such that each expectation exist then prove that E(X1+…….+Xn  ) = E(X1)+……..+E(Xn).      Assume discrete.

Q: 29. Let = {1,...,r}" = {(x₁,...,xn): x; E {1,..., r), i = 1, ..., and assume an assignment of…

A: Given::--  Let = {1,...,r}" = {(x1,...,xn): X; € {1,...,r), i = 1,..., n} and assume an assignment…

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: If X follows the Poisson probability law such that P (x = 1) = P' (x = 2), then ind the probability…

A:

Q: Exercise 3. Let X be a random variable with mean µ and variance o². For a E R, consider the…

A: Farmula used  V(X)=E(X2)-[E(X)]2

Q: (a) f(x) = c(x² + 6), for x = 0, 1, 2, 3 5 3 - (:) ( 2² X (b) f(x) = c 3-x for x = 0, 1, 2

A: It is given that a probability distribution of the discrete random variable X.

Q: . Let Y be a continuous random variable where f(y) F(y)? (a) Y 18 Y 15 Y-3 18 y+3 (b) (c) (d) 18 (e)…

A: The random variable Y is a continuous random variable with pdf f(y) = 1/18 ; -3 ≤ y ≤ 15

Q: E(Y) = a+ bE(X)+ c[E(x)]² + cVar(X) when X is a discrete random variable. You must use the…

A: For discrete distribution, E(X) = Σ x P(x) Var(X) = E(X2) - [E(X)]2 Now, given that Y = a + bX + cX2…

Q: If a random variable X has a discrete uniform distribution. fx(x)=1/k for x=1,2,..,k;0 otherwise.…

A: From the given information, let X be a discrete uniform distribution. The probability mass function…

Q: Let X be a discrete random variable with pmf (a) Find the pmf for Y = X². (b) Find the pmf for U = X…

A: The PMF of random variable X is: f(x)=18      x=-214      x=-114      x=014      x=118       x=2  0…

Q: A discrete random variable X can only take the values -1, 0 and 1. The probabilities of this are P(X…

A:

Q: Let Y₁ 1 X=(x,-X₂), where X, and X₂ are independent random variables and each of them is distributed…

A: Given that the independent random variables X1 and X2 follows χ22 distribution.

Q: Let X be a Poisson random variable with E(X) = 3. Find P(2 < x < 4).

A: When X~ Pois(λ) P(X=x)=e-λ λx/x!  ; X=0,1,2,3,.. result: r!=r(r-1)(r-2)..3x2x1

Q: Random variables X and Y are connected through pgf ¤x(s) and Vy(s) by the relation x(s) = s¥y(/s)…

A:

Q: Consider a random variable X taking the values k1, k2, ·.. , kn E R with probability P1, P2, ; Pn E…

A: Formula is

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: (24) Let X be a random variable with p.d.f. 2e-2x 0<x<00 f(x) = find E(e*) O.W

A: Given,f(x)=2e-2x   ;0&lt;x&lt;∞0           ;o.w

Q: Exercise 3. Let X be a random variable with mean and variance o2. For a € R, consider the…

A: Let X be the  random variable with mean μ and variance σ2. For aεℝ i.e. E(X)=μ and V(X)=σ2 We know…

Q: Let X be a random variable with pmf P(0)=0.2, P(1)-0.5, P(2)=0.3. Let F(X) be the cdf of X. (a) Find…

A: let X be a random variable. PMF of X ; P(0) = 0.2, P(1) = 0.5, P(2) = 0.3 F(X) be the cdf of X.

Q: Let X be one observation from a population with pmf |x| f(x|0) = (9) ¹ (1 – 0)¹−|™|, x = −1, 0, 1; 0…

A:

Q: 4. Let X, Y be independent random variables. Prove that E(XY) = E(X) E(Y) and hence show that V(X…

A: The expected value is also known as average of the probability distribution function. It is the…

Q: Theorem 11. Let X be a random variable and let g(x) be a non-negative function. Then for r > 0,…

A:

Q: 4. Assume that X is an exponential random variable. Suppose further that Var(X) 5- E(X). (a) Find…

A:

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.