a.
Show that
Derive the
a.
Answer to Problem 59SE
It is verified that
The
Explanation of Solution
Given info:
A random sample
Calculation:
Let
The probability density function of the random variable
Probability value:
Therefore, it is verified that
Confidence interval based on given probability statement:
The confidence interval for
Thus, the
b.
Show that
Derive the
b.
Answer to Problem 59SE
It is verified that
The
Explanation of Solution
Calculation:
Let
The probability density function of the random variable
Probability value:
Therefore, it is verified that
Confidence interval:
The confidence interval for
Thus, the
c.
Find the shorter interval among the obtained two intervals.
Find the 95% confidence interval for
c.
Answer to Problem 59SE
The confidence interval obtained in part (b) is shorter than the interval obtained in part (a).
The 95% confidence interval for
Explanation of Solution
Given info:
The data represents the sample of 5 waiting times of a morning bus. The variable waiting time for a morning bus is uniformly distributed.
Calculation:
The confidence interval for
Width of the confidence interval obtained in part (b):
The confidence interval for
Width of the confidence interval obtained in part (b):
Here, the value of
That is,
And the sample size n will be always greater than 1.
That is,
From this it can be said that,
Therefore, the width of the interval in part (b) will be narrower than the width of the interval in part (a).
Thus, the shorter confidence interval for
95% confidence interval:
For 95% confidence level,
Thus, the level of significance is
The maximum waiting time for a sample of 5 waiting times is
The 95% confidence interval for
Thus, the 95% confidence interval for
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Chapter 7 Solutions
Probability and Statistics for Engineering and the Sciences
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- The random variable Y has probability density function f(V) = k(y + y³), 0 2. Hence find PG < Y <). iii) Find the variance of Y.arrow_forwardSuppose that X and Y are continuous random variables with joint pdf given by c(x²+y?) 0arrow_forwardTwo discrete random variables X and Y have the joint probability density function: 2'e *p' (1-p) f(x.y)%3D y!(x-y)! y = 0,1,2,., x; x = 0,1,2,. where 1, p are constants with 2 >0 and 0arrow_forwardLet X1, X2,., Xn be a random sample from a uniform distribution on the interval [0, 0] , so that f(x) = 1/0 if 0 s x< 0 Then if Y = max (X), it can be shown that the random variable U = Y/0 has density function f(u) = nun-1 if 0 sus1 If P( (a/2)1/n < Y/0 < (1-a/2)/n)=1-a а. Derive a 100(1-a)% Cl for 0 based on this probability statement. If my waiting time for a morning bus is uniformly distributed and observed waiting times are x,=4.2, x2=3.5, X3=1.7 ,X4=1.2 , and x5=2.4, (Use 3 digits after decimal point) 95% CI for 0 is [ b. If P( a/n < Y/0 < 1)=1-a Derive a 100(1-a)% CI for 0 based on this probability statement. If my waiting time for a morning bus is uniformly distributed and observed waiting times are x1=4.2, x2=3.5, X3=1.7 ,X4=1.2 , and x5=2.4, (Use 3 digits after decimal point) 95% CI for 0 is [ Which of the two intervals derived previously is shorter? C.arrow_forwardLet X and Y be continuous random variables having a joint pdf given by f(x, y) = e-*, 0sysx 3).arrow_forwardSuppose that the random variables Y1 and Y2 have joint probability density function, f(y1, y2), given 6y y2, 0syı < y2, y1 + y2 < 2 fV1, y2) = - 0, elsewhere Derive the marginal density of Y1. Derive the marginal density of Y2.(Hint: this will have to broken down into two parts!!!) Derive the conditional density of Y2 given Y1 = y1. Find P(Y2 < 1.1 | Y1 = .60). %Darrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_iosRecommended textbooks for you
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