91Ó°ÊÓ

the duration of a particular type of criminal trial is known to be normally distributed with a mean of 21 days and a standard deviation of seven days.

We have to define the random variable$X$.

A random variable is a variable that reflects the probability in a random experiment. Upper case alphabets such as $X,Y,$and so on are used to represent it.

The given case entails keeping track of the length of a specific type of criminal trial. Every trial at a particular interval has a different duration. As a result, it can be thought of as a random variable with the following definition:

$X:$The number of days that a certain sort of criminal trial lasts (in days).

$X$seems to be a continuous random variable because it can take an endless number of values in a given interval.

We have to fill$X~\_\_\_\_\_(\_\_\_\_\_,\_\_\_\_\_)$

$X$is a random variable with a mean of $21$and a standard deviation of $7$that represents the length (in days) of a specific sort of criminal trial. If a random variable $X$has a normal distribution with a mean of $\mathrm{μ}$and a standard deviation of s, it is denoted as $X~N(\mathrm{μ},s)$. As a result, the random variable $X$in this situation is denoted as: $X~N(21,7)$

We need to figure out what the chances are that a randomly picked trial will last at least 24 days.

The probability that a randomly selected trial will last at least 24 days is calculated as follows:

$$\begin{gathered} P(xâ‰\yen 24)=P\left(\frac{x-\mathrm{μ}}{\mathrm{σ}}â‰\yen \frac{24-21}{7}\right) \\ =P(zâ‰\yen 0.4286) \end{gathered}$$

The shaded zone in the graph below depicts the needed probability. As a result, it can be written:

$$\begin{gathered} P(zâ‰\yen 0.4286)=0.5-P(0≤z≤0.4286) \\ =0.5-0.1659 \end{gathered}$$

From normal table : $0.3341$

We must determine the number of days required to complete 60% of criminal cases.

The needed probability is calculated as follows:

$P(x<k)=P\left(\frac{x-\mathrm{μ}}{\mathrm{σ}}<\frac{k-21}{7}\right)=P\left(z<z_1\right)=0.6$

We get the following results from the normal distribution tables:

$z_1=0.2533$

From the equation $x=\mathrm{μ}+z\mathrm{σ}$, we have to replace $x$with $k$and $z$with $z_1$

$$\begin{gathered} k=21+0.2533*7 \\ =22.77 \end{gathered}$$