5. The mgf of a discrete random variable X is given by 1 1 M(t)=ce'|- (2-e²)² (3-e²)²] (a) Find the value of e, the expectation E[X(2X-5)]. t< In 2
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![5. The mgf of a discrete random variable X is given by
M(t)= ce
1
(2-e¹)²
+
1
(3-e¹)²
t< In 2
(a) Find the value of e, the expectation E[X(2X-5)].
(b) Find the pmf of X and P[X≥ 3].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff6d69502-ffbd-48d7-85dd-a2e3610b14b4%2Fcd9d334c-8439-498f-bad2-56e512c19bf7%2F6p9gv6r_processed.jpeg&w=3840&q=75)
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- The conditional probability of E given that F occurs is P(EF)=___________. So in rolling a die the conditional probability of the event E, getting a six, given that the event F, getting an even number, has occurred is P(EF)=___________.4.20. How can I show that X is a Poisson random variable with parameter lambda, then E[Xn] =.... ? And after, using this result to compute E[X3]?Consider two securities, the first having μ1 = 1 and σ1 = 0.1, and the secondhaving μ2 = 0.8 and σ2 = 0.12. Suppose that they are negatively correlated,with ρ = −0.8. Denote the expected return and its standard deviation as functions of π byμ(π ) and σ (π ). The pair (μ(π ), σ (π )) trace out a curve in the plane as πvaries from 0 to 1. Plot this curve in R.
- (b) Let Z be a discrete random variable with E(Z) = 0. Does it necessarily follow that E(Z³) = 0? If yes, give a proof; if no, give a counterexample.3.) Suppose X has probability generating function GX(t) = 0.2 + 0.3t + 0.1t2 + 0.4t3. What is P(X = 2)? What is P(X = 0)?Let X be a Poisson random variable with E(X) = 3. Find P(2 < x < 4).
- X is a discrete random variable and takes the values 0,1 and 2 with probabilities of 1/6, 1/3 and 1/2, respectively. What is the moment generator function M(t) of X?Consider two assets (X and Y) with mX = 10%, mY = 10%, σX2=.16, σY2=.25, and Cov(X,Y) = -.125. What is the expected return and variance of the portfolio having 70% invested in X and 30% invested in Y? Compare the risk and return of this portfolio with the risks and returns associated with investing everything in either X or Y. a) What is p(XY)? b) What is the expected return of the portfolio (m.7X+.3Y)? c) What is the standard deviation of the portfolio (s.7X+.3Y)? d) How does the standard deviation of the portfolio (s.7X+.3Y) compare to the standard deviations of assets X and Y? Show all work and formulas used in EXCELIf the moment generating function of a random variable X is: (1/3+(2/3)e t ) 5 find P (X > 3).