Question

One of the most important functions is called the standard normal curve. it is defined 2T I. Plot a graph of this curve from

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Answer #1

We use R for all the exercises that follow:

1.

I will paste the R code along with the explanation and the output:

Code:

> curve(1/(sqrt(2*pi))*exp(-(x^2)/2),from=-10,to=10,ylab="PDF of standard normal distribution")

Explanation:

The curve() function is used to plot a curve in R, and takes as the first argument the function which needs to be plotted. "from" and "to" arguments specify the x-values between which the graph will be plotted. "ylab" is used to label the y-axis.

Output:

PDF of standard normal distribution 0.0 0.1 0.2 0.3 0.4

2.

Code (with explanation and output)

> #First we generate a sequence of values between 0 and 1 and with step of 0.01. We will get a vector of 101 values.
> x=seq(from=0,to=1,by=0.01)
> #Then we evaluate the value of f(x) at each of these x-values.
> y=1/(sqrt(2*pi))*exp(-(x^2)/2)
> #Then we find the sum of these y-values and subtract half of the first and last y-value.
> z=sum(y)-(y[1]/2)-(y[101]/2)
> #Then we multiply the value obtained by 0.01 to get the answer
> Answer=0.01*z
> #We then round the value to four decimal places
> Ans=round(Answer,digits=4)
> print(Ans)
[1] 0.3413

Thus, we obtain 0.3413 as the answer.

3.

By looking at the graph, we observe that it is symmetric about zero.

Thus, the area under the curve lying between, x=-1 and x=0 is equal to the area under the curve lying between x=0 and x=1. Now, since the are under the curve corresponds to probability, we get :
P(-1 \le Z \le 0) = P(0 \le Z \le 1)

Similarly,

Area to the left of zero is equal to the right of zero.

Thus,

P(Z \le 0) = P(Z \ge 0)

But Z has a probability distribution. Thus,

\begin{align*} P(Z \le 0) + P(Z \ge 0) &= 1 \\ \Rightarrow P(Z \le 0) + P(Z \le 0) &=1 \\ \Rightarrow \ \ \ \ \ \ \ \ \ \ \ \ 2*P(Z \le 0) &= 1 \\ \Rightarrow \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ P(Z \le 0) &= 0.5 \end{align*}

For any queries, feel free to comment and ask.

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