Question

a random number generator picks a number from 4 to 61 in a uniform manner. round answers to 4 decimal places when possible.
a. the mean of this distribution is
b. the standard deviation is
c. the probability that the number will be exactly 59 is p(x = 59) =
d. the probability that the number will be between 11 and 46 is p(11 < x < 46) =
e. the probability that the number will be larger than 21 is p(x > 21) =
f. p(x > 20 | x < 27) =
g. find the 53rd percentile.
h. find the minimum for the upper quartile.
hint:
written hint
helpful videos: probability +, conditional probability + conditional probability + percentiles +

Explanation:

Step1: Recall mean formula for uniform distribution

For a uniform distribution $X\sim U(a,b)$, the mean $\mu=\frac{a + b}{2}$. Here $a = 4$ and $b=61$, so $\mu=\frac{4 + 61}{2}=\frac{65}{2}=32.5$.

Step2: Recall standard - deviation formula for uniform distribution

The standard deviation $\sigma=\sqrt{\frac{(b - a)^2}{12}}$. Substitute $a = 4$ and $b = 61$ into the formula: $\sigma=\sqrt{\frac{(61 - 4)^2}{12}}=\sqrt{\frac{57^2}{12}}=\sqrt{\frac{3249}{12}}\approx16.4317$.

Step3: Probability of a single point in continuous distribution

For a continuous uniform distribution, the probability of a single point is $0$. So $P(X = 59)=0$.

Step4: Calculate $P(11\lt X\lt46)$

The probability density function of $U(a,b)$ is $f(x)=\frac{1}{b - a}$ for $a\leq x\leq b$. Here $f(x)=\frac{1}{61 - 4}=\frac{1}{57}$. Then $P(11\lt X\lt46)=\frac{46 - 11}{61 - 4}=\frac{35}{57}\approx0.6140$.

Step5: Calculate $P(X\gt21)$

$P(X\gt21)=\frac{61 - 21}{61 - 4}=\frac{40}{57}\approx0.7018$.

Step6: Calculate $P(X\gt20|X\lt27)$

By the formula for conditional probability $P(A|B)=\frac{P(A\cap B)}{P(B)}$. Here $A=\{X\gt20\}$ and $B = \{X\lt27\}$, so $A\cap B=\{20\lt X\lt27\}$. Then $P(X\gt20|X\lt27)=\frac{27 - 20}{27 - 4}=\frac{7}{23}\approx0.3043$.

Step7: Calculate the 53rd percentile

Let the 53rd percentile be $k$. For a uniform distribution, $P(X\lt k)=\frac{k - a}{b - a}$. We want $P(X\lt k)=0.53$, so $\frac{k - 4}{61 - 4}=0.53$, then $k-4=0.53\times57$, and $k=4 + 0.53\times57=4+30.21 = 34.21$.

Step8: Calculate the minimum for the upper quartile

The upper quartile is the 75th percentile. Let it be $q$. Using $P(X\lt q)=\frac{q - a}{b - a}$, with $P(X\lt q)=0.75$. Then $\frac{q - 4}{61 - 4}=0.75$, so $q-4=0.75\times57$, and $q=4+0.75\times57=4 + 42.75=46.75$.

Answer:

a. 32.5
b. 16.4317
c. 0
d. 0.6140
e. 0.7018
f. 0.3043
g. 34.21
h. 46.75