04 Random Variables and Probability Distributions Free Online FlashCards

What is a random variable?

A random variable assigns a numerical value to each outcome of a random process; uppercase letters such as $X$ denote the variable.

How do discrete and continuous random variables differ?

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

What conditions must a discrete PMF satisfy?

A valid PMF satisfies $0\leq P(X=x)\leq 1$ for every $x$, and $\sum_xP(X=x)=1$.

For the message distribution, what is $P(X\geq2)$?

Add the probabilities for 2 and 3: $P(X\geq2)=0.30+0.15=0.45$.

When is a binomial distribution appropriate?

A binomial distribution counts successes in a fixed number of independent two-outcome trials with a constant success probability.

How does a continuous PDF represent probability?

For a continuous variable, probability is area under the PDF over an interval; the PDF value at one point is not itself a probability.

What is $P(X=c)$ for a continuous random variable?

For any continuous random variable, the probability of one exact value is $P(X=c)=0$.

What does a CDF measure?

The CDF is $F(x)=P(X\leq x)$, and interval probability is found by $P(a<X<b)=F(b)-F(a)$.

For $X\sim\operatorname{Uniform}(0,10)$, find $P(2<X<6)$.

The interval has width $6-2=4$ and density $\frac{1}{10}$, so $P(2<X<6)=4\left(\frac{1}{10}\right)=0.40$.

What is the expected value of a discrete random variable?

For a discrete variable, $E(X)=\sum_x xP(X=x)$. It is the long-run average value.

What is the expected message count per hour?

Using $E(X)=\sum_xxP(X=x)$, the message distribution has $E(X)=0(0.20)+1(0.35)+2(0.30)+3(0.15)=1.40$.

What does variance measure?

Variance is the probability-weighted average of squared deviations from the mean: $\operatorname{Var}(X)=\sum_x(x-\mu)^2P(X=x)$.