Random Variables: A Comprehensive Guide for Nursing Students

Question 1

What is the probability mass function of a discrete random variable X?

Accepted Answer

P(X = x) for all possible values of x

Answer

E(X)

Answer

Var(X)

Question 2

What is the expected value of a continuous random variable X?

Accepted Answer

The mean of X

Answer

The median of X

Answer

The variance of X

Answer

The mode of X

Question 3

Which of the following probability distributions is used to model the time between events in a Poisson process?

Accepted Answer

Poisson distribution

Answer

Uniform distribution

Answer

Exponential distribution

Answer

Normal distribution

Question 4

Which of the following is NOT a characteristic of a normal distribution?

Accepted Answer

U-shaped

Answer

Unimodal

Answer

Symmetric

Answer

Bell-shaped

Question 5

Which of the following is a key concept in random variable theory?

Accepted Answer

Independence

Answer

Causality

Answer

Correlation

Answer

Regression

Question 6

Which of the following applications of random variables is particularly useful in statistics?

Accepted Answer

Modeling random phenomena

Answer

Generating random numbers

Answer

Predicting future events

Question 7

Which of the following scenarios represents a random variable?

Accepted Answer

The number of patients admitted to a hospital on a specific day

Answer

The average temperature in a specified month

Answer

The height of all adults in a particular zip code

Question 8

In a fair coin flip experiment conducted 10 times, what is the expected value of the number of heads?

Accepted Answer

5

Answer

7.5

Answer

2.5

Answer

10

Question 9

Which of the following random variables is continuous?

Accepted Answer

The duration of a nursing shift

Answer

The number of medications prescribed for a patient

Answer

The number of patients in a hospital ward

Answer

The gender of a patient

Question 10

What is the probability distribution function of a binomial random variable with n = 6 and p = 0.3?

Accepted Answer

P(X = k) = (6 choose k) * (0.3)^k * (0.7)^(6-k)

Answer

P(X = k) = (6 * k) * (0.3)^k * (0.7)^(6-k)

Answer

P(X = k) = (6 * k) * (0.3)^(6-k) * (0.7)^k

Answer

P(X = k) = (6 choose k) * (0.3)^(6-k) * (0.7)^k

Question 11

A Poisson random variable has a mean of 4. What is its variance?

Accepted Answer

4

Answer

2

Answer

1

Answer

8

Question 12

Which of the following properties is unique to the normal distribution?

Accepted Answer

It is symmetric around its mean and has bell-shaped curves

Answer

It is bimodal

Answer

It is skewed to the right

Answer

It is only defined for positive values

Question 13

What is the standard deviation of a uniform random variable with a minimum value of 5 and a maximum value of 10?

Accepted Answer

√5/6

Answer

5/√6

Answer

√5/3

Answer

5/3

Question 14

Which of the following is a measure of the spread or variability of a random variable?

Accepted Answer

Standard deviation

Answer

Mode

Answer

Mean

Answer

Median

Question 15

Which of the following statements about random variables is true?

Accepted Answer

A random variable represents the possible numerical outcomes of an experiment.

Answer

The probability distribution of a random variable cannot be determined.

Answer

A random variable is always a constant.

Answer

A random variable can take on negative values only.

Question 16

What type of random variable can only take on a finite or countable number of values?

Accepted Answer

Discrete

Answer

Categorical

Answer

Continuous

Answer

Mixed

Question 17

What is the expected value of a discrete random variable X with probability mass function P(X = x)?

Accepted Answer

∑[x * P(X = x)]

Answer

∫[x * P(X = x)]

Answer

max[x * P(X = x)]

Question 18

Which of the following is the variance of a continuous random variable X with probability density function f(x)?

Accepted Answer

∫[(x - μ)^2 * f(x)]

Answer

min[(x - μ)^2 * f(x)]

Answer

max[(x - μ)^2 * f(x)]

Answer

∑[(x - μ)^2 * f(x)]

Question 19

What is the probability that a continuous random variable X with a mean of 10 and a standard deviation of 2 will take on a value between 8 and 12?

Accepted Answer

0.6827

Answer

0.1357

Answer

0.9545

Answer

0.0545

Question 20

What is the probability mass function of a discrete random variable X that takes on values 0, 1, and 2 with probabilities 0.4, 0.3, and 0.3, respectively?

Accepted Answer

P(X = 0) = 0.4, P(X = 1) = 0.3, P(X = 2) = 0.3

Answer

P(X = 0) = 0.3, P(X = 1) = 0.3, P(X = 2) = 0.4

Answer

P(X = 0) = 0.3, P(X = 1) = 0.4, P(X = 2) = 0.3

Question 21

A nurse is studying the number of patients admitted to the emergency room per day. What type of random variable is this?

Accepted Answer

Discrete

Answer

Ordinal

Answer

Categorical

Answer

Continuous

Question 22

A study is conducted to examine the effectiveness of a new medication for treating anxiety. The outcome variable is the level of anxiety, measured on a scale from 1 to 10. What type of random variable is the anxiety level?

Accepted Answer

Ordinal

Answer

Nominal

Answer

Discrete

Answer

Continuous

Question 23

A nurse is interested in the probability of a patient developing a post-operative infection. What type of probability distribution would be most appropriate to model this event?

Accepted Answer

Binomial

Answer

Exponential

Answer

Normal

Answer

Poisson

Question 24

A hospital records an average of 3 patients with a particular type of infection per week. What probability distribution would be most appropriate to model the number of patients with this infection in a given week?

Accepted Answer

Poisson

Answer

Normal

Answer

Exponential

Answer

Binomial

Question 25

A nurse is studying the time it takes for a specific medication to reach its peak concentration in the bloodstream. What type of probability distribution would be most appropriate to model this time?

Accepted Answer

Exponential

Answer

Binomial

Answer

Normal

Answer

Poisson

Question 26

What is the expected value of a random variable?

Accepted Answer

The average value of the random variable over a large number of trials.

Answer

The value of the random variable with the highest probability.

Answer

The most likely value of the random variable.

Question 27

What does the variance of a random variable measure?

Accepted Answer

The spread or variability of the random variable.

Answer

The difference between the highest and lowest values of the random variable.

Answer

The average value of the random variable.

Answer

The probability of a specific value of the random variable.

Question 28

A nurse is interested in the average length of stay for patients admitted to a specific hospital unit. She collects data on 100 patients and finds the average length of stay to be 5 days. What is the best estimate of the population mean length of stay?

Accepted Answer

5 days

Answer

Less than 5 days

Answer

It is impossible to determine without further information.

Answer

More than 5 days