Q: A variable $$X is $$uniformly distributed on $$[a, b]. $$What is the probability density function (PDF) of $$X$$?

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

Question 1

A random variable $$X is $$uniformly distributed between $$a = 20$$ and $$b = 50. $$What is the probability that $$X$$ takes a value less than $$30$$?

Accepted Answer

$$P(X \u003c 30) = \frac{30 - 20}{50 - 20} = \frac{10}{30} = 0.333$$

Question 2

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

Accepted Answer

$$\mu = \frac{a + b}{2}$$

Question 3

Suppose $$X is a $$standard normal random variable. What is $$P(X \u003e 2)$$?

Accepted Answer

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

Question 4

For a standard normal variable $$Z$$, what is the probability that $$Z is $$less than $$-1.5$$?

Accepted Answer

$$P(Z \u003c -1.5) \approx 0.0668$$

Question 5

A uniform random variable $$X is $$defined 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 6

If $$X is a $$standard normal variable, what is the value of $$P(0 \u003c X \u003c 1.96)$$?

Accepted Answer

$$P(0 \u003c X \u003c 1.96) \approx 0.4750$$

Question 7

A variable $$X is $$uniformly distributed on $$[a, b]. $$What is the probability density function (PDF) of $$X$$?

Accepted Answer

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

Question 8

A standard normal variable $$Z$$ has mean $$\mu$$ and standard deviation $$\sigma. $$What are the values of $$\mu$$ and $$\sigma$$?

Accepted Answer

$$\mu = 0$$, $$\sigma = 1$$

Question 9

Suppose the heights of adult males are normally distributed with mean $$\mu = 175 cm $$and standard deviation $$\sigma = 8 cm. $$What is the probability that a randomly selected male is taller than 191 cm?

Accepted Answer

$$P(X \u003e 191) = P\left(Z \u003e \frac{191 - 175}{8}\right) = P(Z \u003e 2) \approx 0.0228$$

Question 10

A uniform random variable $$X is $$defined on $$[0, 100]. $$What is the probability that $$X is $$between $$20$$ and $$80$$?

Accepted Answer

$$P(20 \u003c X \u003c 80) = \frac{80 - 20}{100 - 0} = \frac{60}{100} = 0.6$$

Question 11

If $$X is a $$normal random variable with mean $$\mu = 50$$ and standard deviation $$\sigma = 10$$, what is the $$z-$$score for $$X = 65$$?

Accepted Answer

$$z = \frac{65 - 50}{10} = 1.5$$

Question 12

A standard normal variable $$Z$$ has $$P(Z  1.28)$$?

Accepted Answer

$$P(Z \u003e 1.28) = 1 - 0.8997 = 0.1003$$

Question 13

A random variable $$X is $$uniformly distributed on $$[a, b]. $$What is the cumulative distribution function (CDF) for $$X$$?

Accepted Answer

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

Question 14

Suppose $$X is a $$normal random variable with mean $$\mu = 100$$ and standard deviation $$\sigma = 15. $$What is the probability that $$X is $$less than $$85$$?

Accepted Answer

$$P(X \u003c 85) = P\left(Z \u003c \frac{85 - 100}{15}\right) = P(Z \u003c -1) \approx 0.1587$$

Question 15

A standard normal variable $$Z$$ has $$P(Z  0)$$?

Accepted Answer

$$P(Z \u003e 0) = 0.5$$

Question 16

If $$X is a $$uniform random variable on $$[a, b]$$, which of the following is the variance of $$X$$?

Accepted Answer

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

Question 17

A hospital records the time (in minutes) for a blood test, which is uniformly distributed between $$a = 5$$ and $$b = 25. $$What is the probability that a test takes more than $$15$$ minutes?

Accepted Answer

$$P(X \u003e 15) = \frac{25 - 15}{25 - 5} = \frac{10}{20} = 0.5$$

Question 18

A standard normal variable $$Z$$ has $$P(Z  2.33)$$?

Accepted Answer

$$P(Z \u003e 2.33) = 1 - 0.9901 = 0.0099$$

Question 19

Suppose $$X is a $$normal random variable with mean $$\mu = 60$$ and standard deviation $$\sigma = 12. $$What is the probability that $$X is $$between $$60$$ and $$84$$?

Accepted Answer

$$P(60 \u003c X \u003c 84) = P(0 \u003c Z \u003c 2) = 0.4772$$

Probability Distributions: Uniform and Normal Random Variables

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