Question

Exercice 2:1 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: f(x) = -{ cos x if (i < x <T/2 0 elsewhere. 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

Q1) The cumulative distributive function for X here is obtained as:

F (2) = P(X<r

F2(x) = { $(t) at

$F_x(x) = \int_{0}^{x}Cos \ t \ dt$

This is the required CDF here.

The median value here is computed as:

M = Sin-10.5 6

This is the required median value here.

The 65th percentile value here in similar way as:

P0.65 = Sin-10.65 = 0.07177

This is the required 65th percentile value here.

Q2) The percentage here is computed as:

$F(\pi/3) = Sin (\pi/3) = 0.8660$

therfore 86.60% is the required percentage here.

Q3) The mean and variance of the given new 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

This is the required variance here.