The cumulative probabilities for a continuous random variable X are P(X lessthan

Question

The cumulative probabilities for a continuous random variable X are P(X lessthanorequalto 13) = 0.33 and P(X lessthanorequalto 22) = 0.51. Calculate the following probabilities. (Round your answers to 2 decimal places.) For a continuous random variable X, P(27 lessthanorequalto X lessthanorequalto 74) = 0.35 and P(X > 74) = 0.10. Calculate the following probabilities. (Leave no cells blank - be certain to enter "0" wherever required. Round your answers to 2 decimal places.) A random variable X follows the continuous uniform distribution with a lower bound of -10 and an upper bound of 22. a. What is the height of the density function f(x)? (Round your answer to 4 decimal places.) b. What are the mean and the standard deviation for the distribution? (Round your answers to 2 decimal places.) c. Calculate P(X lessthanorequalto -8). (Round intermediate calculations to 4 decimal places and final answer to 4 decimal places.)

Explanation / Answer

Each question answered with detailed calculations:

1

P(X<=13) = .33

P(X<=22) = .51

a. P(X>13) = 1-P(X<=13) = 1-.33 = .67

b. P(X>22) = 1-P(X<=22) = 1-.51 = .49

c. P(13<X<22) = P(X<=22) - P(X<=13) = .51-.33 = .18

2.

P(27<=X<=74) = .35

P(X>74) = .10

P(X<=74) = 1-p(X>74) = 1-.10 = .90

P(X<27) = 1-P(X>=27) = 1- ( P(27<=X<=74) + P(X>74) ) = 1-.35-.10 = .55

a. P(X<74) = 0, since it is not possible to get this value from whatever has been given in the question

b. P(X<27) = .55 ( as calculated above)

c. P(X=74) = 0, since it iss not possible to get this value from whatever has been given in the question

3.

a. The height is f(x)

So, f(x)*(22-(-10)) = 1, since the total probability or the area under curve is 1

f(x)= 1/32

b. Mean of the uniform dist = (b+a)/2 = (22+(-10))/2 = 6

Stdev of the uniform dist = sqrT((b-a)^2/12) = sqrt((22-(-10))^2/12) = sqrt(32^2/12) = 16/sqrt(3)

c. P(X<=-8) = (x-a)/b-a = -8-(-10) / 22-(-10) = 2/32 = 1/16

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