Concept explainers
a.
Prove that
a.
Explanation of Solution
Calculation:
Binomial probability distribution:
A discrete random variable Y is said to follow a Binomial distribution with mean
The mean and variance of Y are
Beta distribution:
A continuous random variable Y is said to follow a Beta distribution with parameters
The mean and variance of Y are
The conditional expectation of any real valued function
It is given that the conditional distribution of Y given p follows Binomial distribution with parameters n and p where p follows Beta distribution with parameters
Thus, the expected value of the conditional distribution of
For two random variables
Hence,
Therefore, it is proved that
b.
Prove that
b.
Explanation of Solution
Calculation:
For two random variables
From Part (a), it is obtained that
Hence,
Therefore, it is proved that
Want to see more full solutions like this?
Chapter 5 Solutions
Mathematical Statistics with Applications
- If X is a random variable with pdf f(x) = 2x − 2 where x = (1, 2), find the variance of Y = 2X - 3.arrow_forwardLet X ~ U[0,1] and Y = -βln(1-X). What is the distribution of Y? Justify.arrow_forwardSuppose X and Y are independent. X has a mean of 1 and variance of 1, Y has a mean of 0, and variance of 2. Let S=X+Y, calculate E(S) and Var(S). Let Z=2Y^2+1/2 X+1 calculate E(Z). Hint: for any random variable X, we have Var(X)=E(X-E(X))^2=E(X^2 )-(E(X))^2, you may want to find EY^2 with this. Calculate cov(S,X). Hint: similarly, we have cov(Z,X)=E(ZX)-E(Z)E(X), Calculate cov(Z,X). Are Z and X independent? Are Z and Y independent? Why? What about mean independence?arrow_forward
- Let X1, . . . , Xn be iid from Exp(β). Find the distribution of the sample maximum Y = X(n). Is this an exponential distribution? (This question was rejected this morning. I did double check and there is no missing info with this question.)arrow_forwardSuppose X and Y are two random variables with E[X] = 1, Var (X) = 4, E[Y ] = -1, Var (Y) 4, and Cov (X, Y) = 1. Find the standard deviation of (X - Y). = (a) 2 (b) √2 (c) 6 (d) √6 (e) None of the abovearrow_forward8. Given the Beta Distribution where p(x) = Calculate the mean and variance 1 B(a, b) x-¹(1-x)b-1 +1 B(a,b) = ¹ x-¹(1 x)b-¹dx = = - r(a) (b) I(a + b)arrow_forward
- Example: Suppose that X u = (120, 80) and covariance matrix (X1, X2) has a bivariate Normal distribution with mean 6. 3 Σ= 5 What are the mean and variance of a' X, where a = (1, –1)T?arrow_forward4. Suppose that X has pdf f(x) = 3x² for 0 < x< 1. Find the pdf of the random variable Y = VX.arrow_forward2.2 Explain how to generate from the Beta (1, 3) distribution using the inverse- transform method.arrow_forward
- In a certain city, the daily consumption of electricpower in millions of kilowatt-hours can be treated as arandom variable having a gamma distribution with α = 3and β = 2. If the power plant of this city has a daily capacity of 12 million kilowatt-hours, what is the prob-ability that this power supply will be inadequate on any given day?arrow_forwardAn investor has found that company1 have an expected return on E(X) = 4% and variance for the return equal V(X) = 0.49. Company 2 has E(Y) = 6% and variance V(Y) = 0.64. The correlation between the companies return is ρ(X,Y) = 0.3. The investor wants to invest p (0<p<1) in company1 and (1-p) in company2. The combined investment have a return: R = pX + (1-p)Y. Let p=0.4 such that R= 0.4X +0.6Y. Find the Expectation and variance of R. how are these results in comparison with X and Y separatley?arrow_forward
- Calculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,