04 Random Variables and Probability Distributions Online Quiz Questions

True or false: A random variable assigns a numerical value to each outcome of a random process.

Which variable is continuous rather than discrete?

A random variable $X$ gives the number of messages received in an hour. The probabilities are $P(X=0)=0.20$, $P(X=1)=0.35$, $P(X=2)=0.30$, and $P(X=3)=0.15$. What is $P(X\geq 2)$?

Using the distribution $P(X=0)=0.20$, $P(X=1)=0.35$, $P(X=2)=0.30$, and $P(X=3)=0.15$, calculate the expected number of messages per hour. Enter the value as a decimal number.

Select all statements that correctly describe a continuous random variable and its probability density function.

True or false: The expected value of a discrete random variable must be one of the values that the random variable can take.

Which situation is most appropriately modeled by a binomial distribution?

For the message distribution $P(X=0)=0.20$, $P(X=1)=0.35$, $P(X=2)=0.30$, and $P(X=3)=0.15$, the mean is $\mu=1.4$. Calculate $\operatorname{Var}(X)$. Enter the numerical value in squared message-count units.

Select all statements that correctly describe transformations of a random variable.

A game pays $\$8$ with probability $0.25$ and costs $\$3$ with probability $0.75$. Calculate the expected net payoff per play and interpret the result in context. Explain why the expected value does not determine the outcome of every individual play.

True or false: The law of large numbers implies that the relative frequency of an outcome tends to approach its theoretical probability as the number of repetitions increases.

Which random variable is continuous?

True or false: For a continuous random variable, including the endpoints does not change the probability of an interval.

A random variable $X$ has probabilities $P(X=0)=0.20$, $P(X=1)=0.35$, $P(X=2)=0.30$, and $P(X=3)=0.15$. What is $P(X\geq 2)$?

What is the name of the function that gives $P(X=x)$ for each possible value of a discrete random variable?

True or false: A table can represent a valid discrete probability distribution only if every probability is between 0 and 1 inclusive and the probabilities sum to 1.

If $Y=3X+7$, which expression gives $\operatorname{Var}(Y)$ in terms of $\operatorname{Var}(X)$?

Which conditions are required for a binomial distribution? Select all that apply.

A bus waiting time $X$ is uniformly distributed on $[0,10]$ minutes. What is $P(2<X<6)$? Enter the probability as a decimal.

What is the name of the function $F(x)=P(X\leq x)$ for a random variable?

A random variable $X$ has standard deviation $4$. If $Y=3X+7$, what is $\operatorname{SD}(Y)$?

If a random variable measures waiting time in minutes, what are the units of its variance?

Which quantity is best modeled as a continuous random variable?

A discrete random variable $X$ has the following proposed probabilities: $P(X=0)=0.25$, $P(X=1)=0.40$, $P(X=2)=0.30$, and $P(X=3)=0.20$. What is the correct assessment of this proposed distribution?

A help desk receives calls independently at an average rate of 12 calls per hour. Which distribution is most appropriate for modeling the number of calls received during one hour?

A continuous random variable $X$ represents an exact measurement such as waiting time. What is $P(X=5)$, assuming the PDF is valid and defined around 5?

A discrete random variable has $P(X=0)=0.20$, $P(X=1)=0.50$, and $P(X=2)=0.30$. What is its expected value $E(X)$?