Question

complete the probability distribution table.
probability distribution table

$x$	$p(x)$

|2|
|1|
|0|
how many practices should we expect leah to attend any given week?
what is the standard deviation?
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Explanation:

Step1: Recall probability distribution property

The sum of all probabilities in a probability - distribution $P(x)$ is 1. But since no other information about the probabilities is given, we assume for a discrete uniform distribution over the three values $x = 0,1,2$ that $P(0)=P(1)=P(2)$. So $P(0)=P(1)=P(2)=\frac{1}{3}$.

Step2: Calculate the expected value $E(X)$

The formula for the expected value of a discrete random variable is $E(X)=\sum_{i}x_{i}P(x_{i})$. Here, $E(X)=0\times P(0)+1\times P(1)+2\times P(2)$. Substituting $P(0) = P(1)=P(2)=\frac{1}{3}$, we get $E(X)=0\times\frac{1}{3}+1\times\frac{1}{3}+2\times\frac{1}{3}=\frac{0 + 1+2}{3}=1$.

Step3: Calculate the variance $Var(X)$

The formula for the variance of a discrete random variable is $Var(X)=\sum_{i}(x_{i}-E(X))^{2}P(x_{i})$. First, for $x = 0$: $(0 - 1)^{2}\times\frac{1}{3}=\frac{1}{3}$, for $x = 1$: $(1 - 1)^{2}\times\frac{1}{3}=0$, for $x = 2$: $(2 - 1)^{2}\times\frac{1}{3}=\frac{1}{3}$. Then $Var(X)=\frac{1}{3}+0+\frac{1}{3}=\frac{2}{3}$.

Step4: Calculate the standard deviation $\sigma$

The standard deviation is the square - root of the variance. So $\sigma=\sqrt{Var(X)}=\sqrt{\frac{2}{3}}\approx0.82$.

Answer:

$P(0)=\frac{1}{3}$, $P(1)=\frac{1}{3}$, $P(2)=\frac{1}{3}$
Expected value: 1
Standard deviation: $\sqrt{\frac{2}{3}}\approx0.82$