WJEC Statistics 1 - Key Facts

Question	Answer
What does it mean if events A and B are mutually exclusive? Also, $P(A\cap B)=?$	Events A and B cannot happen at the same time. $P(A\cap B)=0$
What does it mean if events A and B are independent? Also, $P(A\cap B)=?$	If A happens, this does not affect the probability of B happening (and vice versa). $P(A\cap B)=P(A) \times P(B)$
$P(A\|B)=?$ (there is a rearranged version of this given in the formulae book)	$P(A\|B)=\frac{P(A\cap B)}{P(B)}$
If events A and B are independent, then $P(A\|B)=?$	$P(A\|B)=P(A)$
If events A and B are independent, then $P(B\|A)=?$	$P(B\|A)=P(B)$
The addition law for events A and B is $P(A\cup B)=?$ (given in formulae book)	$P(A\cup B)=P(A)+P(B)-P(A\cap B)$
$P(A')=?$	$P(A')=1-P(A)$ $A'$ is called the complement of $A$ and $P(A')$ is the probability of $A$ not happening
For events A and B that are NOT independent, $P(A\cap B)=?$	$\begin{align} P(A\cap B)&=P(A)\times P(B\|A)\\ & =P(B)\times P(A\|B) \end{align}$
Describe this shaded area using set notation Image: 7fe24a27-4d4c-4394-b761-66a7ff1b140d.PNG (image/PNG)	$A\cap B'$ or $B'\cap A$
What is a sample space?	The set of all the possible outcomes of a random experiment
How many unordered samples of size $r$ can be taken from a collection of $n$ objects?	$nCr=\binom{n}{r}=\frac{n!}{r!(n-r)!}$ make sure you know how to get your calculator to do this
For any discrete random variable $X$ , $\text{E}(aX + b) = ?$	$\text{E}(aX + b) = a\text{E}(X) + b$
For any discrete random variable $X$ , $\text{Var}(aX + b) = ?$	$\text{Var}(aX + b) = a^2 \text{Var}(X)$
For a discrete random variable $X$ taking values $x_i$ with probabilities $p_i$ , $\text{E}(X)=?$ (given in formulae book)	$\text{E}(X)=\sum x_i p_i$
For a discrete random variable $X$ taking values $x_i$ with probabilities $p_i$ , $\text{Var}(X)=?$ (given in formulae book)	$\begin{align} \text{Var}(X)&=\sum x_i^2 p_i -\mu^2\\ &=\text{E}(X^2)-(\text{E}(X))^2 \end{align}$
Describe this shaded area using set notation Image: a0d16dcc-e38b-43b9-9776-72ea3c79b208.PNG (image/PNG)	$A'\cap B$ or $B\cap A'$
Give the formula for the expected value of a function $g(X)$ of a discrete random variable (given in formulae book)	$E[g(X)]=\sum g(x) P(X=x)$
$X \sim B(n,p)$ $\text{E}(X)=?$ (given in formulae book)	For the binomial distribution $X \sim B(n,p)$ , $\text{E}(X)=np$
$X \sim B(n,p)$ $\text{Var}(X)=?$ (given in formulae book)	For the binomial distribution $X \sim B(n,p)$ , $\text{Var}(X)=npq=np(1-p)$
$X \sim Po(\lambda)$ $\text{E}(X)=?$ (given in formulae book)	For the Poisson distribution $X \sim Po(\lambda)$ , $\text{E}(X)=\lambda$
$X \sim Po(\lambda)$ $\text{Var}(X)=?$ (given in formulae book)	For the Poisson distribution $X \sim Po(\lambda)$ , $\text{Var}(X)=\lambda$
Describe this shaded area using set notation Image: 49d67efe-6a8e-4af0-b978-391f68f2f47a.PNG (image/PNG)	$A \cup B$ or $B \cup A$
How would you use the Binomial or Poisson tables to find $P(X=n)$ ?	$P(X=n)=P(X\leq n)-P(X\leq n-1)$
How would you use the Binomial or Poisson tables to find $P(X>n)$ ?	$P(X>n)=1-P(X\leq n)$
How would you use the Binomial or Poisson tables to find $P(X\geq n)$ ?	$P(X\geq n)=1-P(X\leq n-1)$
How would you use the Binomial or Poisson tables to find $P(X<n)$ ?	$P(X<n)=P(X\leq n-1)$
For a continuous probability distribution, how are $f(x)$ and $F(x)$ related?	$f(x)=F'(x)$ $F(x)=P(X\leq x)=\int _{-\infty} ^x f(t) \, \text{d}t$
If $q$ is the lower quartile of a continuous random variable $X$ with cumulative distribution function $F$ , then $F(q)=?$	$F(q)=P(X\leq q)=0.25$
Describe this shaded area using set notation Image: 3531ebf4-07ee-43b7-989a-585f9f5963a5.PNG (image/PNG)	$A \cap B$
If $m$ is the median of a continuous random variable $X$ with cumulative distribution function $F$ , then $F(m)=?$	$F(m)=P(X\leq m)=0.5$
If $Q$ is the upper quartile of a continuous random variable $X$ with cumulative distribution function $F$ , then $F(Q)=?$	$F(Q)=P(X\leq Q)=0.75$
Give the formula for the expected value of a function $g(X)$ of a continuous random variable (given in formulae book)	$E[g(X)]=\int g(x) f(x) \, \text{d}x$
For a binomial distribution $X\sim B(n,p)$ , what is the formula for $P(X=x)$ ? (given in formulae book)	$\binom{n}{x}p^x(1-p)^{n-x}$
For a Poisson distribution $X\sim Po(\lambda)$ , what is the formula for $P(X=x)$ ? (given in formulae book)	$e^{-\lambda}\frac{\lambda^x}{x!}$
Describe this shaded area using set notation in two ways Image: bb689cdf-3e5c-4a4a-9591-b36bba134e26.PNG (image/PNG)	$A'\cap B'$ or $(A\cup B)'$
How is variance related to standard deviation?	$\text{variance}=(\text{stand. dev.})^2$ OR $\text{stand. dev.}=\sqrt{\text{Variance}}$
$X\sim B(n,p)$ What values can $X$ take?	$0, 1, 2, ..., n$
$X\sim Po(\lambda)$ What values can $X$ take?	$0, 1, 2, ...$ (There is no maximum)

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WJEC Statistics 1 - Key Facts

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