4 Random Variables and Probability Distributions Free Online FlashCards

What is a random variable?

A random variable is a numerical function that assigns a number to each outcome of a random experiment.

How do discrete and continuous variables differ?

A discrete random variable has finitely or countably infinitely many possible values, whereas a continuous random variable can take any value in an interval.

What conditions must a valid PMF satisfy?

A valid PMF satisfies $p_X(x)\ge 0$ for every possible $x$, and $\sum_x p_X(x)=1$.

How is a discrete event probability calculated?

For a discrete variable, add the masses for the values in the event: $P(X\in A)=\sum_{x\in A}p_X(x)$.

What conditions must a valid PDF satisfy?

A PDF is nonnegative and integrates to one: $f_X(x)\ge 0$ and $\int_{-\infty}^{\infty}f_X(x)\,dx=1$.

How is continuous interval probability found?

For a continuous variable, interval probability is area under the density: $P(a\le X\le b)=\int_a^b f_X(x)\,dx$.

What does a CDF represent?

The CDF is $F_X(x)=P(X\le x)$, the probability that the random variable is at most $x$.

How does a CDF give an interval probability?

For $a<b$, subtract CDF values: $P(a<X\le b)=F_X(b)-F_X(a)$.

When is the binomial model appropriate?

A binomial variable counts successes in $n$ independent Bernoulli trials with the same success probability $p$.

What does a geometric random variable count?

A geometric variable counts the number of trials needed to obtain the first success.

What process does the Poisson distribution model?

A Poisson variable counts events in a fixed interval when they occur independently at a constant average rate $\lambda$.

What characterizes a uniform distribution?

A uniform variable on $[a,b]$ has equal probabilities for subintervals of equal length and density $f_X(x)=\frac{1}{b-a}$ within the interval.