Probability Functions of Random Variables Quiz

1. A discrete random variable X has probability mass function P(X=k) = c/k² for k=1,2,3. What is the value of c?

π²/6

6/π²

1

1/2

To find the value of c, we need the total probability to equal 1. The probability mass function is P(X=k) = c/k² for k=1, 2, 3. Summing these probabilities gives c(1 + 1/4 + 1/9). The sum equals 49/36, leading to c = 6/π² after solving for c to ensure the total probability sums to 1.

Explanation

To find the value of c, we need the total probability to equal 1. The probability mass function is P(X=k) = c/k² for k=1, 2, 3. Summing these probabilities gives c(1 + 1/4 + 1/9). The sum equals 49/36, leading to c = 6/π² after solving for c to ensure the total probability sums to 1.

2. For a continuous random variable X with PDF f(x) = 2x on [0,1], find P(X > 0.5).

0.25

0.5

0.75

1

To find P(X > 0.5), we need to calculate the area under the PDF f(x) = 2x from 0.5 to 1. The integral of f(x) from 0.5 to 1 gives us the probability. Evaluating this integral results in 0.75, indicating that there is a 75% chance that the random variable X is greater than 0.5.

Explanation

To find P(X > 0.5), we need to calculate the area under the PDF f(x) = 2x from 0.5 to 1. The integral of f(x) from 0.5 to 1 gives us the probability. Evaluating this integral results in 0.75, indicating that there is a 75% chance that the random variable X is greater than 0.5.

3. The cumulative distribution function F(x) = P(X ≤ x) is always ____.

The cumulative distribution function (CDF) F(x) represents the probability that a random variable X takes on a value less than or equal to x. As x increases, the probability cannot decrease; it either stays the same or increases. Therefore, F(x) is always non-decreasing, reflecting the accumulation of probability as x increases.

Explanation

The cumulative distribution function (CDF) F(x) represents the probability that a random variable X takes on a value less than or equal to x. As x increases, the probability cannot decrease; it either stays the same or increases. Therefore, F(x) is always non-decreasing, reflecting the accumulation of probability as x increases.

Submit

4. If X is uniformly distributed on [a,b], what is E[X]?

(a+b)/2

(b-a)/2

A+b

(a²+b²)/2

For a uniformly distributed random variable X on the interval [a, b], the expected value E[X] represents the average of the endpoints. The formula for the mean of a uniform distribution is derived by taking the midpoint of the interval, calculated as (a + b) / 2. This reflects the central tendency of the distribution.

Explanation

For a uniformly distributed random variable X on the interval [a, b], the expected value E[X] represents the average of the endpoints. The formula for the mean of a uniform distribution is derived by taking the midpoint of the interval, calculated as (a + b) / 2. This reflects the central tendency of the distribution.

5. Which property must a function satisfy to be a valid PDF?

F(x) ≥ 0 and ∫f(x)dx = 1

F(x) > 0 for all x

F(x) is continuous

F(x) is strictly increasing

A valid probability density function (PDF) must satisfy two key properties: it must be non-negative for all possible values of x (f(x) ≥ 0) to ensure probabilities are not negative, and the total area under the curve, represented by the integral of f(x) over its entire range, must equal 1 to reflect the total probability.

Explanation

A valid probability density function (PDF) must satisfy two key properties: it must be non-negative for all possible values of x (f(x) ≥ 0) to ensure probabilities are not negative, and the total area under the curve, represented by the integral of f(x) over its entire range, must equal 1 to reflect the total probability.

6. For a random variable X, Var(X) = E[X²] - (E[X])² is the definition of ____.

Variance quantifies the spread of a random variable around its mean. The formula Var(X) = E[X²] - (E[X])² shows that variance is the expected value of the squared deviations from the mean. It captures how much the values of X differ from the expected value, providing insight into the variability of the data.

Explanation

Variance quantifies the spread of a random variable around its mean. The formula Var(X) = E[X²] - (E[X])² shows that variance is the expected value of the squared deviations from the mean. It captures how much the values of X differ from the expected value, providing insight into the variability of the data.

Submit

7. If X ~ Binomial(n=10, p=0.3), what is E[X]?

3

2.1

7

10

For a binomial distribution, the expected value E[X] is calculated using the formula E[X] = n * p. Here, n is the number of trials (10) and p is the probability of success (0.3). Thus, E[X] = 10 * 0.3 = 3, indicating the average number of successes in 10 trials.

Explanation

For a binomial distribution, the expected value E[X] is calculated using the formula E[X] = n * p. Here, n is the number of trials (10) and p is the probability of success (0.3). Thus, E[X] = 10 * 0.3 = 3, indicating the average number of successes in 10 trials.

8. The standard normal distribution has mean μ = 0 and standard deviation σ = ____.

In a standard normal distribution, the mean is always 0, and the standard deviation is always 1. This specific configuration allows for the normalization of data, enabling comparisons across different datasets by transforming them into a common scale. Thus, the standard deviation is defined as 1 in this context.

Explanation

In a standard normal distribution, the mean is always 0, and the standard deviation is always 1. This specific configuration allows for the normalization of data, enabling comparisons across different datasets by transforming them into a common scale. Thus, the standard deviation is defined as 1 in this context.

Submit

9. For an exponential random variable with rate λ, what is the PDF?

F(x) = λe^(-λx) for x ≥ 0

F(x) = e^(-x) for x ≥ 0

F(x) = λx e^(-λx)

F(x) = 1/λ for x ≥ 0

The probability density function (PDF) of an exponential random variable describes the likelihood of different outcomes. For a rate λ, the PDF is defined as f(x) = λe^(-λx) for x ≥ 0, indicating that the probability decreases exponentially as x increases, with λ controlling the rate of decay.

Explanation

The probability density function (PDF) of an exponential random variable describes the likelihood of different outcomes. For a rate λ, the PDF is defined as f(x) = λe^(-λx) for x ≥ 0, indicating that the probability decreases exponentially as x increases, with λ controlling the rate of decay.

10. If X and Y are independent random variables, what is Cov(X,Y)?

0

E[X]E[Y]

Var(X)Var(Y)

E[XY] - E[X]

When X and Y are independent random variables, the covariance between them is zero. This is because independence implies that the occurrence of X does not affect the occurrence of Y, leading to the formula Cov(X,Y) = E[XY] - E[X]E[Y] simplifying to zero, as E[XY] equals E[X]E[Y].

Explanation

When X and Y are independent random variables, the covariance between them is zero. This is because independence implies that the occurrence of X does not affect the occurrence of Y, leading to the formula Cov(X,Y) = E[XY] - E[X]E[Y] simplifying to zero, as E[XY] equals E[X]E[Y].

11. The moment generating function M_X(t) of a random variable X is defined as E[e^(tX)] and is useful for finding ____ of X.

The moment generating function M_X(t) provides a compact way to calculate all moments of a random variable X. By differentiating M_X(t) with respect to t and evaluating at t=0, one can obtain the moments of X, such as the mean and variance, making it a powerful tool in probability and statistics.

Explanation

The moment generating function M_X(t) provides a compact way to calculate all moments of a random variable X. By differentiating M_X(t) with respect to t and evaluating at t=0, one can obtain the moments of X, such as the mean and variance, making it a powerful tool in probability and statistics.

Submit

12. For a Poisson random variable with parameter λ, both the mean and variance equal ____.

In a Poisson distribution, the parameter λ represents both the average rate of occurrence and the expected number of events in a fixed interval. Consequently, for a Poisson random variable, the mean and variance are both equal to λ, reflecting the unique property of this distribution where these two measures coincide.

Explanation

In a Poisson distribution, the parameter λ represents both the average rate of occurrence and the expected number of events in a fixed interval. Consequently, for a Poisson random variable, the mean and variance are both equal to λ, reflecting the unique property of this distribution where these two measures coincide.

Submit