Concept explainers
a.
Prove that the expected values of all odd integer powers of Z are zero.
That is,
a.
Answer to Problem 199SE
It is proved that
Explanation of Solution
The expected value of
Define the
The function
That is,
Therefore, the expected values of all odd integer powers of Z are zero.
That is,
b.
Show that
b.
Answer to Problem 199SE
It is proved that
Explanation of Solution
The expected value of
Define the function
Since the function
Hence, it is proved that
c.
Find the values of
c.
Answer to Problem 199SE
The values of
Explanation of Solution
The result in Part (b) is as follows:
The values of
Put
Put
Substitute
Substitute
d.
Show that
d.
Explanation of Solution
From result in Part (b).
It is known that
Hence,
Substitute
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Chapter 4 Solutions
Mathematical Statistics with Applications
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- A statistic is denoted by T(X1,.., X„) or simply T(X) .... 14.- Let X1, X, be a random sample from the population with distribution N (0, 1). For each of the following items, obtain the distributions of the T(X) statistics. a) T(X) = (X2 – X1)//2. b) T(X) (X1 + X2) /(X2 – X1)² c) T(X) = X{/X? d) T(X) = (X2+ X1)//(X1 – X2)² 19arrow_forward4. At a particular gas station, gasoline is stocked in a bulk tank each week. Let random variable X denote the proportion of the tank's capacity that is stocked in a given week, and let y denote the proportion of the tank's capacity that is sold in the same week. The joint pdf of X and Y is given by f(x,y) = {³x (3x 0 if 0 ≤ yarrow_forwardLet the random variable X have the moment generating function M(t) = e³t 1-t² = -1 < t < 1 What are the mean and the variance of X, respectively?arrow_forward(a) Let X be a random variable with finite variance. LetY = aX + B for some numbers a, B e R. Compute l1 = E[(Y – E[Y|X])²]. (b) Again, let X be a random variable with finite variance. Additionally, let A ~ U(-1,1) such that X and A are independent and consider Y = AX. You may use without proof that A? 1 X². Compute l2 = E [(Y – E[Y|X])²] expressing your final result in terms of EX*] for some value or values of k that you should specify. (c) [TYPE:] Provide an interpretation of the quantities lį and l2 obtained in parts (a) and (b) above in terms of the ability to predict Y given the value of X and explain any difference you observe. !!arrow_forwardHelp asap!arrow_forwardSuppose X is a random variable, whose pdf is defined as follows: 2x = (²x) (u(x) - u(x − 3)) where u(x) is the unit step function. Determine the conditional pdf fx(x 1arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_iosRecommended textbooks for you
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