2.4 Probability Distributions

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Question 3

The random variable $T$ represents the water temperature level (in $∘C^{\circ}\text{C}$ ) at a specific marine research site. The probability distribution for $T$ is given in the following table:

$t$	2	5	8	10
$P (T = t)$	0.2	0.3	0.3	0.2

Show that $E (T) = 6.3$ .

[2]

Find $V a r (T)$ .

[3]

The random variable $S$ represents the salinity index at the same site, recorded as a discrete index. The probability distribution for $S$ is given in the following table:

$s$	3	6	8	$k$
$P (S = s)$	0.25	0.25	0.25	0.25

Name the probability distribution of $S$ .

[1]

Given that $E (S) = E (T)$ , find the value of $k$ .

[2]

A specific species of bioluminescent jellyfish is found at a depth $D$ meters, where $\sim N(\mu, \sigma^2)$ . In a simulation, two researchers, Alice and Bob, each generate values for the parameters of the distribution. They each obtain a value for $T$ to use as $μ\mu$ and a value for $S$ to use as $σ\sigma$ . For each researcher, the values of $T$ and $S$ are generated independently according to the given distributions. The researcher whose parameters yield the higher probability $P (D > 7.0)$ wins.

Alice obtained $t = 8$ and $s = 3$ . Bob obtained $s = 6$ . Find the probability that Bob wins.

[4]

Determine the largest value of $P (D > 7.0)$ achievable in this simulation.

[2]

Find the probability of a researcher achieving this maximum value.

[1]

t

P (T = t)

0.2

0.3

0.2

s

k

P (S = s)

0.25

2.4 Probability Distributions

Question 3

Markscheme

2.4 Probability Distributions

Question 3

Markscheme

2.4 Probability Distributions

Question 3

2.4 Probability Distributions Questions

2.4 Probability Distributions

Question 3

2.4 Probability Distributions Questions