The expected value, E(X), of a uniform random variable X defined over the interval a

Q: Let X be a random variable with mean u. Then, the variance of X, denoted by Var (X). is defined as…

A: Given that  X be a random variable with mean  To find the variance

Q: Find the expected value of q. d. Let X1, X2,., Xn be i.i.d. Bernoulli(p). Suppose q =

A: Bernoulli Trail: In probability theory, a Bernoulli trial is a random experiment which has exactly…

Q: Consider random variables X and Y with x = 0 x = 1 fx,y(x, y) in the table. Find fxy (1|0) and…

A: Let X and Y be represent the random variable having joint probability distribution f(x,y)  We have…

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: Select one: а. О b. 2 C. var(x) d. - 2 e. 1

A: To find the value E[X(1-X)].

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: The cumulative dietribution function for a random variable X ie given by: (1-e* F(x)={ for x20…

A:

Q: Let X, Y are random variables (r.v.) and a, b, c, d are values. Complete the formulas using the…

A: Step 1: Step 2: Step 3: Step 4:

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: Let X1, X2, X3, X4 be independent random variables of size 4 from a gamma distribution G(a, X) : -…

A: We want to find the expected value E(Z)=X1+X2+X3+X4

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: Suppose that V and W are random variables with Cov(V, W) = 0. Which of the following must also be…

A: Solution : V and W are random variable  If V and W are independent  so E(VW) = E(V).E(W) Cov(VW) =…

Q: lom variable X is given below. fx(x) 1 2 a random variable Y, where whose PDF is given below. fy(y)

A: *Answer:

Q: . Then Var(X+Y+Z) is equal to:

A: Here x , y and z are independent variables  So covariance between them are zero

Q: The probability distribution of random variable X is a) b) c) d) Show that k = 6. Find P(X <3). P(X=…

A: Hi! Thank you for the question. As per the honor code, we are allowed to answer three sub-parts at a…

Q: Consider the sample space S= {(-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4)}, where each point is assumed…

A: Given sample space is, S=(-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4). Each point is equally likely.…

Q: Property 4. If X is a random variable and a and b are constants, then E (a X + b) = a E (X) + b,…

A:

Q: Let X and Y be discrete random variables with joint probability function: P( x, y ) = ( 2x + y )/12,…

A: Let X and Y be discrete random variables with joint probability function:P( x, y ) = ( 2x + y )/12,…

Q: If X: Poisson() and Y: Poisson(3) are independent random variabl Then A = Select one: а. 4 b. 16 C.…

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: A9. Let X and Y be independent standard normally distributed N(0, 1) random variables. What is…

A: Given that  X and Y are independent standard normal distribution N(0 , 1).

Q: For any two random variables Y₁ and Y₂, Var(Y₁Y₂|Y₁ = y₁) = y{Var(Y₂|Y₁ = y₁). TRUE

A: Given that Y1 and Y2 are two random variables. Formula: If X is a random variable, then VX=EX2-EX2.…

Q: Let X and Y be independent random variables. Which statements are true? Select one or more:   a.…

A: We know that,  If X and Y are two independent random variables,  Then,  These Statments are true :…

Q: Let X and Y be two independent random variables such that E[X]=3, E(Y)=4, V(X)=10 and V(Y)-20. Then…

A: According to the given information, we have E(X)=3E(Y)=4V(X)=10V(Y)=20

Q: The cdf of a random variable Y is |1– exp(-0.4y), y 20 F,(y) otherwise Valuate P{2.5 <Y < 6.2}.

A:

Q: Given that X and Y are independent random variables and E(x) = 4, E(Y) = 7, Var(x) = 3, Var(Y) = 6;…

A: x and y are independent events. E(X) = 4 E(Y) = 7 Var(X) = 3 Var(Y) = 6   Var(X-6) = ?

Q: Suppose a random variable X has the moment generating function my (t) = 1/√1 – 2t fo t < 1/2. What…

A: Note: According to Bartleby expert guidelines, we can answer only first question with their three…

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

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Q: Suppose U = 2T - 1 where T is a geometric random variable with parameter p. What is Var(U)?

A:

Q: Let X be a random variable taking only the values 0, 1, 2, 3, 4 with probabilities given by P(X = 0)…

A: Discrete probabilty  The conditions of the probability is 1) p(x) ≥ 0 2) Σp(x)=1

Q: Show that for two random variables X and Y, 2 2 var(aX+bY) = a¹o²+b²oy² + 2ab Cxx XY where a and b…

A:

Q: If X is a continuous random variable that takes on values between 10 and 40, then the P(X = 15.5) =…

A: We have given that  X is a continuous random variable that takes on values between 10 and 40, then…

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 X be a random variable such that E(X – 2)² = 10 and E(X+ 1)² = 4, then 1 а. Е(X) = - b. E(X) = =…

A: How to obtain E(X) 1. Simplify both expression to get two equation 2. Simplify both equation and…

Q: The probability that an event A occurs is 0.1. When A occurs, X is a discrete random variable with…

A:

Q: fx(x) 4 a -1a

A: We have to find the value of constant for valid pdf.

Q: Let X and Y be two random variables. Suppose we know E(X) = 7 and E(Y)=3. Let Z = X − Y be a random…

A: Given EX=7 and EY=3.Let Z=X-Y.Then,E(Z)=4

Continuous Probability Distributions

Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.

Normal Distribution

Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!