Q: Which of the following best describes the probability density function (PDF) of a uniform random variable $$X on$$ $$[a, b]$$?

$$f(x) = \frac{1}{b - a}$$ \text{ for } a \leq x \leq b$

Q: A random variable $$X is $$uniformly distributed on $$[0, 100]. $$What is the median of $$X$$?

$$\text{Median} = \frac{0 + 100}{2} = 50$$

Question 1

Suppose $$X is a $$continuous random variable uniformly distributed on the interval $$[20, 50]. $$What is the probability that $$X$$ takes a value greater than $$35$$?

Accepted Answer

$$P(X \u003e 35) = \frac{50 - 35}{50 - 20} = \frac{15}{30} = 0.5$$

Question 2

Given a uniform random variable $$X on$$ $$[a, b]$$, what is the formula for its mean (expected value)?

Accepted Answer

$$E[X] = \frac{a + b}{2}$$

Question 3

A standard normal random variable $$Z$$ has mean $$0$$ and standard deviation $$1. $$What is $$P(Z \u003e 2)$$?

Accepted Answer

$$P(Z \u003e 2) \approx 0.0228$$

Question 4

If $$X is $$uniformly distributed on $$[10, 40]$$, what is the standard deviation of $$X$$?

Accepted Answer

$$\sigma = \frac{40 - 10}{\sqrt{12}} = \frac{30}{\sqrt{12}} \approx 8.66$$

Question 5

A hospital records the systolic blood pressure of patients, which is normally distributed with mean $$120$$ and standard deviation $$15. $$What is the probability that a randomly selected patient has a systolic blood pressure less than $$105$$?

Accepted Answer

$$P(X \u003c 105) = P\left(Z \u003c \frac{105 - 120}{15}\right) = P(Z \u003c -1) \approx 0.1587$$

Question 6

For a standard normal variable $$Z$$, what is the value of $$P(-1 \u003c Z \u003c 1)$$?

Accepted Answer

$$P(-1 \u003c Z \u003c 1) \approx 0.6826$$

Question 7

A chemical process yields a product whose weight $$X is $$uniformly distributed between $$25$$ and $$35$$ grams. What is the probability that a randomly selected product weighs between $$28$$ and $$32$$ grams?

Accepted Answer

$$P(28 \u003c X \u003c 32) = \frac{32 - 28}{35 - 25} = \frac{4}{10} = 0.4$$

Question 8

If $$X is a $$standard normal random variable, what is the $$z-$$score for a value $$x = 1.5$$?

Accepted Answer

$$z = \frac{x - \mu}{\sigma} = \frac{1.5 - 0}{1} = 1.5$$

Question 9

A university entrance exam score is normally distributed with mean $$70$$ and standard deviation $$10. $$What is the probability that a randomly selected student scores above $$85$$?

Accepted Answer

$$P(X \u003e 85) = P\left(Z \u003e \frac{85 - 70}{10}\right) = P(Z \u003e 1.5) \approx 0.0668$$

Question 10

Which of the following best describes the probability density function (PDF) of a uniform random variable $$X on$$ $$[a, b]$$?

Accepted Answer

$$f(x) = \frac{1}{b - a}$$ 	ext{ for } a \leq x \leq b$

Question 11

A random variable $$X is $$uniformly distributed on $$[0, 100]. $$What is the median of $$X$$?

Accepted Answer

$$	ext{Median} = \frac{0 + 100}{2} = 50$$

Question 12

A standard normal random variable $$Z is $$used to model IQ scores. What is the probability that a randomly selected person has a $$Z-$$score less than $$-2$$?

Accepted Answer

$$P(Z \u003c -2) \approx 0.0228$$

Question 13

If $$X is $$uniformly distributed on $$[a, b]$$, which of the following is the variance of $$X$$?

Accepted Answer

$$	ext{Var}(X) = \frac{(b - a)^2$}{12}$$

Question 14

A machine produces metal rods whose lengths are normally distributed with mean $$50 cm $$and standard deviation $$2 cm. $$What percentage of rods are expected to be longer than $$54 cm$$?

Accepted Answer

$$P(X \u003e 54) = P\left(Z \u003e \frac{54 - 50}{2}\right) = P(Z \u003e 2) \approx 2.28\%$$

Question 15

A random variable $$X is $$uniformly distributed on $$[5, 25]. $$What is $$P(X \u003c 10)$$?

Accepted Answer

$$P(X \u003c 10) = \frac{10 - 5}{25 - 5} = \frac{5}{20} = 0.25$$

Question 16

A standard normal random variable $$Z is $$used to model heights. What is the probability that $$Z is $$between $$-2$$ and $$2$$?

Accepted Answer

$$P(-2 \u003c Z \u003c 2) \approx 0.9545$$

Question 17

A random variable $$X is $$uniformly distributed on $$[30, 80]. $$What is the probability that $$X is $$less than $$50$$?

Accepted Answer

$$P(X \u003c 50) = \frac{50 - 30}{80 - 30} = \frac{20}{50} = 0.4$$

Question 18

A standard normal random variable $$Z is $$used to model test scores. What is the probability that $$Z is $$greater than $$1.96$$?

Accepted Answer

$$P(Z \u003e 1.96) \approx 0.025$$

Probability Distributions: Uniform and Normal Random Variables

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