Question #185804

1. A discrete random variable ๐‘‹ has the following probability mass function

๐‘ƒ(๐‘‹=๐‘ฅ)={2๐‘˜๐‘ฅ ๐‘ฅ=2,4,6

{๐‘˜(๐‘ฅ+2) ๐‘ฅ=8

{ 0 ๐‘œ๐‘กโ„Ž๐‘’๐‘Ÿ๐‘ค๐‘–๐‘ ๐‘’

where ๐‘˜ is a constant

a. Show that ๐‘˜=1/34

b. Find the exact value of ๐‘ƒ(4<๐‘ฅโ‰ค8)

c. Find the exact value of ๐‘ƒ(2<๐‘ฅ<4)

d. What is the expected value of the random variable ๐‘‹?

e. What is the variance of the random variable ๐‘‹?

f. Determine ๐‘‰๐‘Ž๐‘Ÿ(5โˆ’3๐‘‹)


1
Expert's answer
2021-04-27T17:07:07-0400

P(X=x)={2kx,โ€‰โ€‰x=2,4,6k(x+2),โ€‰โ€‰x=80,โ€‰โ€‰otherwiseP(X = x) = \left\{ {\begin{matrix} {2kx,\,\,x = 2,4,6}\\ {k(x + 2),\,\,x = 8}\\ {0,\,\,otherwise} \end{matrix}} \right.

Find the values โ€‹โ€‹of the probabilities

P(x=2)=2kโ‹…2=4kP(x = 2) = 2k \cdot 2 = 4k

P(x=4)=2kโ‹…4=8kP(x = 4) = 2k \cdot 4 = 8k

P(x=6)=2kโ‹…6=12kP(x = 6) = 2k \cdot 6 = 12k

P(x=8)=kโ‹…(8+2)=10kP(x = 8) = k \cdot (8 + 2) = 10k

a) We find the value of k from the condition

โˆ‘Pi=1\sum {{P_i}} = 1

Then

4k+8k+12k+10k=1โ‡’34k=1โ‡’k=1344k + 8k + 12k + 10k = 1 \Rightarrow 34k = 1 \Rightarrow k = \frac{1}{{34}}

Q.E.D

b) construct a series of probability distributions

X2468P43483412341034\begin{matrix} X&2&4&6&8\\ P&{\frac{4}{{34}}}&{\frac{8}{{34}}}&{\frac{{12}}{{34}}}&{\frac{{10}}{{34}}} \end{matrix}

Let's find

P(4<xโ‰ค8)=P(6)+P(8)=12+1034=1117P(4 < x \le 8) = P(6) + P(8) = \frac{{12 + 10}}{{34}} = \frac{{11}}{{17}}

Answer: P(4<xโ‰ค8)=1117P(4 < x \le 8) = \frac{{11}}{{17}}

c) Let's find

P(2<X<4)=0P(2 < X < 4) = 0 (since, by condition, P(X=x)=0P(X = x) = 0 for all xโˆˆ(2;4)x \in \left( {2;4} \right) )

Answer: P(2<X<4)=0P(2 < X < 4) = 0

d) Let's find he expected value:

M(x)=โˆ‘xipi=2โ‹…4+4โ‹…8+6โ‹…12+8โ‹…1034=9617M(x) = \sum {{x_i}} {p_i} = \frac{{2 \cdot 4 + 4 \cdot 8 + 6 \cdot 12 + 8 \cdot 10}}{{34}} = \frac{{96}}{{17}}

Answer: M(x)=9617M(x) = \frac{{96}}{{17}}

e) Let's find the variance:

Var(x)=M(x2)โˆ’M2(x)=4โ‹…4+16โ‹…8+36โ‹…12+64โ‹…1034โˆ’(9617)2=1120289Var(x) = M({x^2}) - {M^2}(x) = \frac{{4 \cdot 4 + 16 \cdot 8 + 36 \cdot 12 + 64 \cdot 10}}{{34}} - {\left( {\frac{{96}}{{17}}} \right)^2} = \frac{{1120}}{{289}}

Answer: Var(x)=1120289Var(x) = \frac{{1120}}{{289}}

f) Let's find

Var(5โˆ’3x)=Var(5)+Var(โˆ’3x)=0+(โˆ’3)2Var(x)=9โ‹…1120289=10080289Var(5 - 3x) = Var(5) + Var( - 3x) = 0 + {( - 3)^2}Var(x) = 9 \cdot \frac{{1120}}{{289}} = \frac{{{\rm{10080}}}}{{289}}



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