(b) Consider a binomial random variable X with parameters n and p. Let Y = X/n be a new random. Show that the expected value of Y is p and the variance of Y is pq/n. If p(y) is the p.m. f. for Y, show that nY~b(n,p). What are the possible values that Y can take on?

Q: If the variances for independent random variables X and Y is Var(X)=0.15 and Var(Y)=5.62 then Var…

A: Given that  Var(X)=0.15 Var(Y)=5.62

Q: If X and Y are independent random variables with variances Var(X) = 5 and Var(Y)= (a) find the…

A: Given  V(x)=5, V(y)=3,  Z=-2x+4y-3  X and Y are independent

Q: A random variable Y has a uniform distribution over the interval (?1, ?2). Derive the variance of Y.…

A:

Q: Let X be a continuous random variable with mean μ and standard deviation σ. If X is transformed to Y…

A:

Q: X is a discrete uniform (1,3) random variable. When X=x, Y is discrete uniform (1,x). (a) Find…

A: We are given here that:P(X = 1) = P(X = 2) = P(X = 3) = 1/3 as X is a discrete uniform distribution…

Q: (b) Consider a binomial random variable X with parameters and p. Let Y = X/n be a new random. Show…

A: Given that X~b(n,p)So, E(X)=np and Var(X)=npqGiven that Y=XnSo, E(Y)=EXn=1nE(X)=1n(np)=p…

Q: If the random variable X and Y have the joint probability distribution f(x, y) = x+y, 0<x< 1,0 <y<1…

A:

Q: Let the random variable Y = a + bX. Then, µy = bux. O True O False

A: Given statement isY=a+bX

Q: 3. Let X1 and X2 be uncorrelated random variables. Let Y₁ = X₁ + X2 and Y2 = X1 - X2. (a) Find…

A: and  are uncorrelated random variables.

Q: U is a uniform (0, 1) random variable and X = −ln(1 − U). Find the PDF, CDF of random variable X.…

A: Here, it is provided that a random variable U has uniform distribution over 0,1. Another random…

Q: A large group of mice is kept in a cage having compartments A, B, and C. Mice in compartment A move…

A:

Q: Let X and Y be independent random variables distributed as N(-1, 1) and N(1, 1), respectively. Find…

A: X follows a  normal distribution with , µ= -1 , σ= 1. Y follows a  normal distribution with , µ= 1 ,…

Q: If X is a binomially distributed radom variable with E(X) var (X) 4/3; find the distribution of X. 2…

A:

Q: (a) A lower bound for P(6<Y <24) (b) the value of k such that P(|Y-15| ≥k) ≤ 0.09.

A:

Q: Let X and Y be independent standard normal random variables. Determine the pdf of W = X² + y² using…

A:

Q: (i) the maximum value of p.d.f. (ii) expectation of X, E(X) (iii) variance of X.

A:

Q: we are evaluating two random variables X and Y with probabilitydistribution equal to: p(x, y) = P (X…

A:

Q: b. Find E[X(X – 1)] for Poisson random variable. Use the result to find the variance of X.

A:

Q: f X is a uniformly distributed random varibale with a=9 and b=16, then Calculate the…

A: Answer: From the given data, X follows uniform distribution, a = 9   and   b = 16

Q: What is the distribution of X? 1) Out of the 38 pizzas that you try, how many would you expect to…

A: Let p be the probability of you liking the pizza. Given that p = 0.08

Q: Let X be a discrete random variable. (i) Prove that V ar(X) = E(X2) ? (EX)2. Hint: Use the…

A: We want to prove both results

Q: (Math 161A/163 review) Suppose that X₁, X2, X3 are independent Exponential (A) random variables. A>…

A:

Q: Let x be a discrete Probability distribution Fi Cand x= -L.0,l,2+3 į Find the value of K i. Find the…

A: Discrete Random Variable  Two numbers are often used to summarize a probability distribution for a…

Q: Let x be a dis crete random variable with probability distribution fCx) =k (x² + 20) and x =…

A: i. From the given information, the probability distribution function is f(x) = K (X2 + 20) and x=…

Q: Random variables X and Y have the means and standard deviations as given in the table to the right…

A:

Q: Let X be a geometric distribution function with px(n) = p(1 - p)n-¹. Find the expected value and the…

A: P(X=x)=p(1−p)x−1;x=1,2,3,...

Q: Let x be random variable with E(x) = 2 and var(x) = 3. Verify that random variable x and the random…

A:

Q: Let X and Y be random variables with variances Var(X) = 1 and Var(Y ) = 2. (Note that X and Y might…

A: It is given that   Var(X) = 1 Var(Y) = 2

Q: Three zero mean, unit variance random variables X, Y, and 2 are added to form a new random variable,…

A: Random variables and  have zero mean and unit variances.That is:  and These variables are added to a…

Q: û = ax + by, where a and b are real numbers. Find the expected value of this estimator in terms of a…

A:

Q: continuous random variables Z is known to have an Erlang (2,1.3) type PDF. what is the variance of Z

A: Given that Z is a continuous random variable having a distribution Erlang with parameters (…

Q: Let X, Y N (2, 3) two normal random variables with mean 2 and variance 3. Which of the following…

A: Normal distribution  mean , median and mode are eqaul

Q: Compute the correlation between the random variables Y1 and Y5 = 1+ X1 + X2.

A:

Q: Let X be the Random Variable: x | P(X = x) 0.5 0 1 0.2 2 0.3 Let X1, X2 be the identically…

A:

Q: Let x1  and x1 be independent standard normal random variables. If Y1= x1 + 2x2 and  Y2 = 2x1 - x2,…

A: We know, any linear combination of normal distribution is a normal distribution. Here, x1 and x2 are…

Q: QUESTION: Let X have the density f(r) = 2r if 0 < a <1 and f(x) = 0 otherwise. Show that X has the…

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Q: A random variable X has a distribution with probability function f(x) = k for x = 1, 2, ... (a) Find…

A:

Q: Let w₁, W2,..., wt be independent random variables from a normal distribution with mean 0 and…

A: Let w1, w2......wt be independent random variables from a normal distribution with mean 0 and…

Q: Let X represent the number that occurs when a 5-sided red die is tossed and Y the number that occurs…

A: a)The random variable X defines the number that occurs when a 5-sided red die is tossed.

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.