Victorian VCE · VCE Mathematical Methods

Probability, Binomial and Normal Distributions

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The random variable $X$ has a binomial distribution $X \sim Bi (n = 10, p = 0.2)$ . The standard deviation of $X$ is:
Let $Z \sim N (0, 1)$ . Given that $Pr (Z > 1.2) = 0.1151$ , find $Pr (- 1.2 < Z < 1.2)$ .
A discrete random variable $Y$ has probability function $Pr (Y = y) = k (y + 1)$ for $y \in {0, 1, 2, 3}$ . Find the exact value of $k$ .
Given that events $A$ and $B$ are independent with $Pr (A) = 0.4$ and $Pr (B) = 0.5$ , find $Pr (A^{'} \cup B^{'})$ .
The weights of newborn puppies are normally distributed with mean $500 g$ and standard deviation $40 g$ . If $10%$ of puppies are classified as underweight, find the threshold weight to…
A continuous random variable $X$ has probability density function $f (x) = \frac{3}{8} x^{2}$ for $0 \leq x \leq 2$ , and $0$ elsewhere. Calculate the exact variance $Var (X)$ .
$X \sim Bi (n = 5, p = 0.4)$ . Find $Pr (X = 2)$ to four decimal places.
$X \sim Bi (n = 6, p = 0.5)$ . Find $Pr (X \geq 4 ∣ X \geq 1)$ .
$X \sim N (μ, σ^{2})$ with $Pr (X < 10) = 0.0228$ and $Pr (X < 16) = 0.5$ . Using $Pr (Z < - 2) = 0.0228$ , the values of $μ$ and $σ$ are:
$f (x) = k x (2 - x)$ for $0 \leq x \leq 2$ and $0$ elsewhere is a probability density function. The value of $k$ is:
Events $A$ and $B$ are mutually exclusive with $Pr (A) = 0.3$ and $Pr (B) = 0.5$ . Then $Pr (A \cup B) =$
For the binomial random variable $X \sim Bi (n = 20, p = 0.3)$ , the mean and variance are:
$X \sim Bi (n = 4, p = 0.25)$ . The probability $Pr (X \geq 1)$ , to four decimal places, is:
The marks on a test are normally distributed with mean 70 and standard deviation 8. Given $Pr (Z < 2) = 0.9772$ , the probability that a mark exceeds 86 is:
For events $A$ and $B$ : $Pr (A) = 0.6$ , $Pr (B) = 0.5$ and $Pr (A \cap B) = 0.3$ . Then $Pr (A ∣ B) =$
A continuous random variable $X$ has probability density function $f (x) = 2 x$ for $0 \leq x \leq 1$ , and 0 elsewhere. Then $Pr (X \leq 0.5) =$
A fair coin is tossed 6 times. Given that at least 2 heads occurred, the probability of exactly 3 heads is:
$X \sim N (50, σ^{2})$ and $Pr (X > 62) = 0.0668$ . Given $Pr (Z < 1.5) = 0.9332$ , the value of $σ$ is:
A discrete random variable $X$ has $Pr (X = x) = k x$ for $x = 1, 2, 3, 4$ . The expected value $E (X)$ is:
Each trial of an experiment succeeds with probability 0.1, independently. The smallest number of trials $n$ for which the probability of at least one success exceeds 0.9 is:

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Probability, Binomial and Normal Distributions

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