QuestionQ15: For $X \sim B(n, p)$ we have $\operatorname{Var}(X)=0.8 E(X)$ . Find $p$

Studdy Solution

STEP 1

1. $X \sim B(n, p)$ denotes a binomial distribution with parameters $n$ (number of trials) and $p$ (probability of success).
2. The variance of a binomial distribution is given by $\operatorname{Var}(X) = np(1-p)$ .
3. The expected value of a binomial distribution is given by $E(X) = np$ .
4. We are given that $\operatorname{Var}(X) = 0.8 E(X)$ .

STEP 2

1. Express the variance and expected value in terms of $n$ and $p$ .
2. Set up the equation based on the given relationship between variance and expected value.
3. Solve the equation for $p$ .

STEP 3

Express the variance and expected value in terms of $n$ and $p$ :
$\operatorname{Var}(X) = np(1-p)$
$E(X) = np$

STEP 4

Set up the equation based on the given relationship $\operatorname{Var}(X) = 0.8 E(X)$ :
$np(1-p) = 0.8(np)$

STEP 5

Solve the equation for $p$ :
First, divide both sides by $np$ (assuming $np \neq 0$ ):
$1 - p = 0.8$
Subtract 0.8 from both sides:
$1 - 0.8 = p$
$p = 0.2$
The value of $p$ is:
$\boxed{0.2}$

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