Question

Exercice 2: A biostatistician is interested in studying the time X (in seconds) it takes a hematology cell counter to complete a test on a blood sample. The probability density function of the aforementioned time is: COST f(x) = = {* if 0 < x < 1/2 elsewhere. 0 1. Find the cumulative distribution function and compute the median and 65-th percentile of the time to complete a test on a sample. 2. What is the percentage of tests require less than 7/3 seconds to complete? 3. Suppose that this biostatistician is interested in the following composite function of time to complete a test on a sample: exp(sin x). Compute the mean and variance of the underlying random variable.

Answer 1

a) The cumulative distribution function for X here is computed as:

$F_x(x) = P(X \leq x)$

F (2) = cost dt

This is the required cumulative distribution function for X here.

F.(20) = Sin x, ,0<x</2

The median here is computed as:

FM) = 0.5

$M = Sin^{-1}0.5 = \frac{\pi}{6}$
This is the required median here.

The 65th percentile value now is computed in similar way as:

$P_{0.65} = Sin^{-1}0.65 = 0.0717\pi$

This is the required 65th percentile value here.

Q2) The percentage of tests less than seconds is computed here as:

고

$F(\pi/3) = Sin(\pi/3) = 0.8660$
Therefore 86.60% is the required percentage here.

Q3) The mean and variance of the given random variable here are computed as:

-1/2 Elesin 1) = $* (Cos thesin dx = lesin 1177 =e-1

Therefore (e - 1) = 1.7183 is the required mean value here.

Now we first compute the second moment here as:

e2sin 1/2 e2-1 E(e2sin 1) = **/2 (Cos x)e2sin I de ." = 2 2 0

Now the variance here is computed as:

1 Var(esin 1) = E(e2S int) – [E(esint)]2 - (e - 1) 2

Var(esin) e? - 1 - 2e? – 2 + 4e 2

-e Var(esin T) = - 3+ 4e 2

Var(esin 1) = -0.5e2 – 1.5 + 2e

Therefore 0.2420 is the required variance here.