Show that al(w) is a random variable, where EEF and a is a constant. Calculate E[X] using the basic definition of an expectation as done in class.

Q: Prove - If X and Y are independent random variables, then E(XY ) = E(X)E(Y ). (You can assume that…

A: Assume that X and Y are two independent discrete random variables.

Q: Let X be a nonnegative random variable. Prove that. EX (EX) 4E[X*] 2

A: Given that X be a nonnegative random variable. Result: PX≥c≤E(X2)c2               ≥c2EX2

Q: 4.14 Show that if X and Y are independent random variables, then var[X + Y] = var[X] + var[Y]. You…

A: If  and  are independent random variables, then we have to show thatalso we have to remember that if…

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: Let X be a continuous random variable symmetric about Y. Let Z = 1 if X>Y OR Z = 0 if X <= Y. Find…

A: Given: Z=1 if X&gt;Y OR Z=0 if X&lt;=Y

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: Compute the conditional expectation E(Y|X = 0.4). Are X and Y independent? Explain.

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Q: Consider generalizing in the following way: N customers line up for their sandwiches, and you are…

A: The probability of an event is the chances of happening of the event. It can be defined as the ratio…

Q: the probability

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Q: Suppose that X and Y are discrete random variables on the probability space (Q.F.P). Show that E[aX]…

A: The objective of this question is to prove two properties of the expected value (E) of random…

Q: be a random variable with the followi Value x of X P(X=x) -1 0.25 0.35 0.15 0.15 0.10 plete the…

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Q: Given an event A C S, define a random variable X by: { 1 1 if s E A, A. 0 if s X(s) = =

A: Given information: It is given that the random variable X is an indicator variable and X(s)…

Q: Let X be the number of students who show up for a professor's office hour on a particular day.…

A: Probability of an event is computed by taking the ratio of number of favourable outcomes to total…

Q: Which of the following statements are true? a. Probability density functions are always less or…

A: Solution Probability density function is always non negative and less than or equal to 1.…

Q: X is a discrete random variable with p.m.f given by P(X=x)= x/c where c is a constant. The support…

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Q: Assume that X is a random variable with probability mass function summarized in the table below. x…

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Q: If X is a uniform random variable over the interval (8, 12), find E[X]

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Q: Let X denote the number of empty bins when 60 balls are thrown independently into 30 bins. What is…

A: Let X denote the number of empty bins. Balls are thrown independently into bins. The number of balls…

Q: Let E and F be independent events such that P(E) = 0.4, P(F) = 0.6.. Find P(E| F) and P(E U F). 5.

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Q: E be the event that a new car requires engine work under warranty and let T be the event that the…

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Q: Let Z be a discrete random variable withE(Z)=0. Does it necessarily follow that E(Z3)=0? If yes,…

A: Solution

Q: The following figure shows probability distributions, namely, Expont and Expon2, of inter-arrival…

A: In this case, two exponential distributions are given.

Q: (c) Compare and contrast the probabilities you got in part (c). Which one has largest probability?…

A: Given that from part (c): P (-0.5&lt;X&lt;0.5) =0.35667 P (-0.5&lt;Y&lt;0.5) =0.382375 P…