Questions: Question If X ~ U(4.5,18.5) is a continuous uniform random variable, what is P(X<11) ? Select the correct answer below: 1/28 3/28 13/28 9/14 19/28

Question If X ~ U(4.5,18.5) is a continuous uniform random variable, what is P(X<11) ?

Select the correct answer below: 1/28 3/28 13/28 9/14 19/28
Transcript text: Question If $X \sim U(4.5,18.5)$ is a continuous uniform random variable, what is $P(X<11)$ ? Select the correct answer below: $\frac{1}{28}$ $\frac{3}{28}$ $\frac{13}{28}$ $\frac{9}{14}$ $\frac{19}{28}$ FEEDBACK MORE INSTRUCTION SUBMIT Content attribution
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Solution

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Solution Steps

Step 1: Calculate the Mean

The mean E(X) E(X) of a uniform distribution U(a,b) U(a, b) is given by the formula:

E(X)=a+b2 E(X) = \frac{a + b}{2}

Substituting the values a=4.5 a = 4.5 and b=18.5 b = 18.5 :

E(X)=4.5+18.52=232=11.5 E(X) = \frac{4.5 + 18.5}{2} = \frac{23}{2} = 11.5

Step 2: Calculate the Variance

The variance Var(X) \text{Var}(X) of a uniform distribution is calculated using the formula:

Var(X)=(ba)212 \text{Var}(X) = \frac{(b - a)^2}{12}

Substituting the values:

Var(X)=(18.54.5)212=(14)212=19612=16.3333 \text{Var}(X) = \frac{(18.5 - 4.5)^2}{12} = \frac{(14)^2}{12} = \frac{196}{12} = 16.3333

Step 3: Calculate the Standard Deviation

The standard deviation σ(X) \sigma(X) is the square root of the variance:

σ(X)=Var(X)=16.33334.0415 \sigma(X) = \sqrt{\text{Var}(X)} = \sqrt{16.3333} \approx 4.0415

Step 4: Calculate the Probability P(X<11) P(X < 11)

The cumulative distribution function F(x;a,b) F(x; a, b) for a uniform distribution is given by:

F(x;a,b)=xaba,axb F(x; a, b) = \frac{x - a}{b - a}, \quad a \leq x \leq b

To find P(X<11) P(X < 11) , we calculate:

P(4.5X11)=F(11)F(4.5) P(4.5 \leq X \leq 11) = F(11) - F(4.5)

Calculating F(11) F(11) :

F(11)=114.518.54.5=6.5140.4643 F(11) = \frac{11 - 4.5}{18.5 - 4.5} = \frac{6.5}{14} \approx 0.4643

And since F(4.5)=0 F(4.5) = 0 :

P(4.5X11)=0.46430=0.4643 P(4.5 \leq X \leq 11) = 0.4643 - 0 = 0.4643

Final Answer

The probability P(X<11) P(X < 11) is approximately 0.4643 0.4643 .

Thus, the answer is:

P(X<11)=1328 \boxed{P(X < 11) = \frac{13}{28}}

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