Answer :
Answer:
[tex]\mu_{x}=np[/tex]
[tex]\sigma^2_{X}=np(1-p)[/tex]
Step-by-step explanation:
∵ When x is a random variable having distribution B(n, p), then for sufficiently large value of n, the following random variable has a standard normal distribution,
[tex]\frac{x-\mu}{\sigma}\sim N(0,1)[/tex]
Where,
[tex]\mu=np[/tex]
[tex]\sigma^2=np(1-p)[/tex]
Here the variable X has a binomial distribution,
Such that, np (1 - p) ≥ 10 ⇒ n is sufficiently large.
Where, n is the total numbers of trials, p is success in each trials,
So, the mean of variable X is,
[tex]\mu_{X} = np[/tex]
And, variance of variable X is,
[tex]\sigma^2{X}=np(1-p)[/tex]
the binomial random variable X is approximately normal with [tex]\rm \mu _x = np[/tex] and [tex]\rm \sigma ^2_x = np[/tex]
What is normal a distribution?
It is also called the Gaussian Distribution. It is the most important continuous probability distribution. The curve looks like a bell, so it is also called a bell curve.
Let x be the random variable having distribution B(n, p), then for a sufficiently large value of n, the following random variable has a standard normal distribution,
[tex]\rm \dfrac{x - \mu }{ \sigma} \sim N(0, 1)[/tex]
Where,
[tex]\rm \mu = np\\\\\sigma ^2 = np(1-p)[/tex]
Here the variable x has a binomial distribution, such that,
[tex]\rm np (1 - p) \geq 10 \Rightarrow n[/tex] is sufficiently large.
Where n is the total number of trials, p is success in each trial.
So, the mean of the variable x is.
[tex]\rm \mu x = np[/tex]
And, the variance of variable x is,
[tex]\rm \sigma ^2 x = np[/tex]
More about the normal distribution link is given below.
https://brainly.com/question/12421652