Recall the definition of the probability density.
We now introduce the concept of a uniform probability distribution:
Definition 2
A distribution is called uniform if, on an interval containing all possible values of a random variable, the distribution density is constant, that is:
Picture 1.
Find the value of the constant $\ C$ using the following distribution density property: $\int\limits^(+\infty )_(-\infty )(\varphi \left(x\right)dx)=1$
\[\int\limits^(+\infty )_(-\infty )(\varphi \left(x\right)dx)=\int\limits^a_(-\infty )(0dx)+\int\limits ^b_a(Cdx)+\int\limits^(+\infty )_b(0dx)=0+Cb-Ca+0=C(b-a)\] \ \
Thus, the uniform distribution density function has the form:
Figure 2.
The graph has the following form (Fig. 1):
Figure 3. Density of uniform probability distribution
Uniform Probability Distribution Function
Let us now find the distribution function for a uniform distribution.
To do this, we will use the following formula: $F\left(x\right)=\int\limits^x_(-\infty )(\varphi (x)dx)$
- For $x ≤ a$, according to the formula, we get:
- For $a
- For $x> 2$, according to the formula, we get:
Thus, the distribution function has the form:
Figure 4
The graph has the following form (Fig. 2):
Figure 5. Uniform probability distribution function.
Probability of a random variable falling into the interval $((\mathbf \alpha ),(\mathbf \beta ))$ under a uniform probability distribution
To find the probability of a random variable falling into the interval $(\alpha ,\beta)$ with a uniform probability distribution, we will use the following formula:
Expected value:
Standard deviation:
Examples of solving the problem for a uniform distribution of probabilities
Example 1
The interval between trolleybuses is 9 minutes.
Compile the distribution function and distribution density of the random variable $X$ waiting for the trolley bus passengers.
Find the probability that the passenger will wait for the trolleybus in less than three minutes.
Find the probability that the passenger will wait for the trolleybus in at least 4 minutes.
Find the mathematical expectation, variance and standard deviation
- Since the continuous random variable $X$ of waiting for the trolleybus is uniformly distributed, then $a=0,\ b=9$.
Thus, the distribution density, according to the formula of the density function of the uniform probability distribution, has the form:
Figure 6
According to the formula of the uniform probability distribution function, in our case, the distribution function has the form:
Figure 7
- This question can be reformulated as follows: find the probability that a random variable of a uniform distribution falls into the interval $\left(6,9\right).$
We get:
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Thus, the uniform distribution density function has the form:
Figure 2.
The graph has the following form (Fig. 1):
Figure 3. Density of uniform probability distribution
Uniform Probability Distribution Function
Let us now find the distribution function for a uniform distribution.
To do this, we will use the following formula: $F\left(x\right)=\int\limits^x_(-\infty )(\varphi (x)dx)$
- For $x ≤ a$, according to the formula, we get:
- For $a
- For $x> 2$, according to the formula, we get:
Thus, the distribution function has the form:
Figure 4
The graph has the following form (Fig. 2):
Figure 5. Uniform probability distribution function.
Probability of a random variable falling into the interval $((\mathbf \alpha ),(\mathbf \beta ))$ under a uniform probability distribution
To find the probability of a random variable falling into the interval $(\alpha ,\beta)$ with a uniform probability distribution, we will use the following formula:
Expected value:
Standard deviation:
Examples of solving the problem for a uniform distribution of probabilities
Example 1
The interval between trolleybuses is 9 minutes.
Compile the distribution function and distribution density of the random variable $X$ waiting for the trolley bus passengers.
Find the probability that the passenger will wait for the trolleybus in less than three minutes.
Find the probability that the passenger will wait for the trolleybus in at least 4 minutes.
Find the mathematical expectation, variance and standard deviation
- Since the continuous random variable $X$ of waiting for the trolleybus is uniformly distributed, then $a=0,\ b=9$.
Thus, the distribution density, according to the formula of the density function of the uniform probability distribution, has the form:
Figure 6
According to the formula of the uniform probability distribution function, in our case, the distribution function has the form:
Figure 7
- This question can be reformulated as follows: find the probability that a random variable of a uniform distribution falls into the interval $\left(6,9\right).$
We get:
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