Question

The binomial probability distribution is a family of probability distributions with every single distribution depending on the values of n and p. Assume that x is a binomial random variable with n = 4.

1. Determine a value of p such that the probability distribution of x is symmetric.
2. Determine a value of p such that the probability distribution of x is skewed to the right.
3. Determine a value of p such that the probability distribution of x is skewed to the left.
4. Graph each of the binomial distributions you obtained in parts a, b, and c. Locate the mean for each distribution on its graph.\
5. In general, for what values of p will a binomial distribution be symmetric? Skewed to the right? Skewed to the left?

Short answer

1. The probability distribution of x is symmetric if p=0.5.
2. The probability distribution of x is skewed to the right if 0.5<p<1.
3. The probability distribution of x is skewed to the left if 0<p<0.5.

e. A binomialdistribution will be symmetric if=0.5, postivelyskewedif 0.5<p<1, and negatively skewedif0<p<0.5.

Step-by-step solution

Definition of a symmetric binomial probability distribution 

a.

In a symmetric binomial probability distribution, the probability of successes will be equal to that of the failures. In the case of tossing a coin, the probabilitiesof heads and tails are equal, and this can be an example of a symmetric binomial probability distribution

Determination of the value 

From the definition, as it can be envisaged,the probability will be the same for both variables, and the calculation is shown below:

$\begin{array}{rcl}\text{Probability}& =& \frac{Total probability}{2}\\ & =& \frac{1}{2}\\ & =& 0.5\\ & & \end{array}$

From the above calculation, it can be envisaged that the probability is 0.5.

Determination of the skewness 

b. Skewness, in terms of a binomial probability distribution, determines which side of the curve has deviated from the normal distribution. Whenever the skewness remains onthe right, it indicates that the skewness is positive, and when the skewness is toward the left, it indicates that the skewness is negative.

Determination of the valueof x 

Whenever the skewness remains positive (to the right), it means that the value of p must be more than 0.5. As the maximum probability can be upto 1, the value will stay between 0.5 and 1.

Determination of the value of p 

c. Whenever the skewness remains negative (to the left), it means that the value of p must be less than 0.5.As the minimum probability can be 0, the value will stay between 0 and 0.5

Elucidation of the graphs obtained in Parta

d. On the horizontal axis, the probabilities are plotted, and on the horizontal axis, the values are plotted from 1 to 5.In this case, it is symmetric.So, the mean value, 3,shows the highest probability

Elucidation of the graphsobtained in Part b

On the horizontal axis, the probabilities are plotted, and on the horizontal axis, the values are plotted from 1 to 8. In this case, it is positively skewed.So, the mean value, 4,doesnot show the highest probability, but instead, 3 showsthe highest probability.

Elucidation of the graphs obtained in Part c

On the horizontal axis, the probabilities are plotted, and on the horizontal axis, the values are plotted from 1 to 8.In this case, it is negatively skewed.So, the mean value, 4,doesnot show the highest probability, but instead, 6 showsthe highest probability.

Elucidation of p when the distribution is symmetric

Here, the value of p remains 0.5, and it indicates that the mean value of x shows the highest probability.In this case, the left tail remains exactly equal to the right tail when plotted on the graph.

Elucidation of p when the distribution is positively skewed

Here, the value of p remains above 0.5, and it indicates that the mean value of x does not show the highest probability.In this case, the right tail remains greater than the left tail when plotted on the graph.

Elucidation of p when the distribution is negatively skewed

Here, the value of p remains below 0.5, and it indicates that the mean value of x does not show the highest probability.In this case, the left tail remains greater than the right tail when plotted on the graph.