Comprehensive Assessment Of Random Variables And Probability Distributions Practice Tests

Question 1

A random variable $X$ represents the number of defective microchips in a batch of $4$. Which of the following best describes the nature of $X$?

Accepted Answer

Discrete random variable

Answer

Continuous random variable

Answer

Deterministic variable

Answer

Sample space variable

Question 2

For a function $f(x)$ to be a valid probability mass function (PMF) for a discrete random variable, it must satisfy $\sum_{x} f(x) = 1$ and $f(x) \geq 0$ for all $x$.

Accepted Answer

True

Answer

False

Question 3

Consider a function $f(x) = c(x^2)$ for $x \in \{1, 2, 3\}$. Determine the value of the constant $c$ such that $f(x)$ is a valid PMF.

Accepted Answer

1/14

Question 4

In a clinical trial, the probability of a patient responding to a treatment is $p = 0.2$. If $10$ patients are treated, what is the probability that exactly $2$ respond, assuming independence?

Accepted Answer

$\binom{10}{2} (0.2)^2 (0.8)^8$

Answer

$\binom{10}{2} (0.2)^8 (0.8)^2$

Answer

$\binom{10}{8} (0.2)^2 (0.8)^2$

Answer

$\frac{10!}{2!} (0.2)^2 (0.8)^8$

Question 5

A factory experiences an average of $\lambda = 3$ machine failures per week. Using the Poisson distribution, calculate the probability of exactly $0$ failures in a given week. (Use $e^{-3} \approx 0.0498$)

Accepted Answer

0.0498

Question 6

Which of the following statements are true regarding the Binomial distribution with parameters $n$ and $p$?

Answer

The mean is $\mu = np$

Answer

The variance is $\sigma^2 = np(1-p)$

Answer

The distribution is always symmetric regardless of $p$

Answer

The variance is $\sigma^2 = np$

Question 7

A Poisson random variable $X$ has a mean $\lambda = 4$. Determine the variance of $X$.

Accepted Answer

4

Question 8

A quality control engineer inspects a batch of 10 components. The probability that a component is defective is $p = 0.05$. Calculate the probability that exactly 2 components are defective using the Binomial distribution formula $P(X=x) = \binom{n}{x} p^x (1-p)^{n-x}$.

Accepted Answer

0.0746

Answer

0.0521

Answer

0.1250

Answer

0.0963

Question 9

A call center receives an average of $\lambda = 3$ calls per minute. Assuming the arrival of calls follows a Poisson distribution, calculate the probability that exactly 5 calls are received in a specific minute. Provide the result rounded to four decimal places.

Accepted Answer

0.1008

Question 10

For a Binomial distribution with $n = 20$ and $p = 0.4$, the variance $\sigma^2$ is equal to $4.8$.

Accepted Answer

True

Answer

False

Question 11

Which of the following statements are true regarding the Poisson distribution with parameter $\lambda$?

Answer

The mean is equal to $\lambda$.

Answer

The variance is equal to $\lambda$.

Answer

The distribution is always symmetric.

Answer

The probability of zero events is $\lambda$.

Question 12

A manufacturing process produces items with a defect rate of $p = 0.01$. In a sample of $n = 100$ items, determine the expected number of defective items $\mu$.

Accepted Answer

1

Answer

0.1

Answer

10

Answer

0.01

Question 13

If the number of accidents at a busy intersection follows a Poisson distribution with a mean of $\lambda = 2$ per week, calculate the probability of having no accidents in a given week. Provide the answer rounded to four decimal places.

Accepted Answer

0.1353

Question 14

For a normal distribution with mean $\mu$ and standard deviation $\sigma$, the probability $P(\mu - 2\sigma < X < \mu + 2\sigma)$ is approximately $0.95$.

Accepted Answer

True

Answer

False

Question 15

If a continuous random variable $X$ follows a normal distribution with $\mu = 50$ and $\sigma = 10$, what is the $Z$-score for $X = 65$?

Accepted Answer

1.5

Answer

0.5

Answer

1.0

Answer

2.0

Question 16

Given a normal distribution $X \sim N(100, 25)$, determine the value of $X$ that corresponds to a $Z$-score of $-1.2$.

Accepted Answer

94

Question 17

A process produces parts with lengths normally distributed with $\mu = 20	ext{ cm}$ and $\sigma = 2	ext{ cm}$. What is the probability that a randomly selected part has a length greater than $24	ext{ cm}$?

Accepted Answer

Option A

Answer

Option B

Answer

Option C

Answer

Option D

Question 18

Which of the following statements are true regarding the standard normal distribution ($Z \sim N(0, 1)$)?

Answer

The mean is 0.

Answer

The standard deviation is 1.

Answer

The curve is symmetric about $Z = 0$.

Answer

The total area under the curve is 0.5.

Question 19

For a normal distribution $X \sim N(75, 10)$, calculate the probability $P(65 < X < 85)$.

Accepted Answer

0.6826

Question 20

If $X \sim N(\mu, \sigma^2)$, what is the probability $P(X < \mu)$?

Accepted Answer

Option A

Answer

Option B

Answer

Option C

Answer

Option D

Comprehensive Assessment Of Random Variables And Probability Distributions Practice Tests

Practice Tests

Comprehensive Assessment of Random Variables and Probability Distributions

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