Use the following to answer questions 1 - 4: Suppose a pdf of X is defined by the function1. What is the probability of ...

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MATH 3339-What is the probability of X

April 14, 2018

Use the following to answer questions 1 - 4: Suppose a pdf of X is defined by the function

1. What is the probability of X < 0.3?
a. 0.3
b. 0.75
c. 0.25
d. 0.5
e. 0.12
2. Determine the probability P(0.275 ? X ? 0.35).
a. 0.075
b. 0.1875
c. 0.8125
d. 0.1375
e. 0.8625
3. Give the first quartile (Q1) of X.
a. 0.25
b. 0.1
c. 2.5
d. 0.3
e. 0.2
4. Find the median of the density curve.
a. 1
b. 0.2
c. 2.5
d. 0.4
e. none of these

For problem 5 - 7, let X have the following pdf.

5.Determine the value of k.
a. 0
b. 1
c. 2
d. 3
e. 0.5

6. Determine the mean of X.
a. 1.5
b. 2.25
c. 3
d. 1.25
e. None of these above.
7. Determine the variance of X.
a. 2.25
b. 5.4
c. 0.3375
d. 0.581
e. None of these above.
Use the following to answer questions 8 - 11. Suppose that a random variable X has a cdf of

8. What is the probability P(X < 0)?
a. 1/9
b. 1/12
c. 2/9
d. 0
e. None of these above.
9. What is the probabiltiy P(0 < X ? 1)?
a. 0.1111
b. 0
c. 0.1389
d. 0.5000
e. None of these above.
10. Find the value of c such that P(X ? c) = 0.1.
a. -0.4642
b. 0.1
c. 1.282
d. 1.922
e. -1.282
11. Determine E[X].
a. 1.5183
b. 1.25
c. 1.50
d. 0
e. None of these above.

12. *Let X denote the amount of time for which a book on 2-hour reserve at a college library is
checked out by a randomly selected student and suppose that X has density function:

Calculate the following probabilities:
a. P(X ? 1)
b. P(0.5 ? X ? 1.5)
c. P(1.5 < X)
13. **The cdf of checkout duration X as described in the previous exercise is

Use this to compute the following:
a. P(X ? 1)
b. P(0.5 ? X ? 1.5)
c. P(1.5 < X)
d. Determine the median checkout duration. That is find x such that F(x) = 0.5.
e. Compute F’(x) to obtain the density function f(x).
14. ***Reconsider the distribution of checkout duration X described in the previous two
problems. Compute the following:

a. E(X)
b. If the borrower is charged an amount h(X) = X2 when checkout duration is X,
compute the expected charge E[h(X)].
c. V(X) and SD(X).
15. For 0 ? x ? 1 let f(x) = kx(1 ? x), where k is a constant.
a. Find the value of k such that f is a density function.
b. Find the cumulative distribution for the previous density function.
16. Suppose the reaction temperature X (in °C) in a certain chemical process has a uniform
distribution by X ~ Unif(-5, 5).

a.
b.
c.
d.

Give the pdf of X.
Compute P(X < 0).
Compute P(-2.5 < X < 2.5)
For k satisfying -5 < k < k +4 < 5, compute P(k < X < k +4).

17. Suppose jobs arrive every 15 seconds on average, ? = 4 per minute, has an exponential
distribution.

a. What is the probability of waiting less than or equal to 30 seconds, i.e .5 min?
b. Compute E(X).
c. Compute Var(X) and SD(X).










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